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<p id="mathjaxlink" class="pcenter"><a href="chap50_mj.html">[MathJax on]</a></p>
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<div class="ChapSects"><a href="chap50.html#X81B00B667D2BD022">50 <span class="Heading">Group Libraries</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X839981CC7D9B671B">50.1 <span class="Heading">Basic Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8489BECB78664847">50.1-1 TrivialGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7A7C473D87B31F3B">50.1-2 CyclicGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X81CCC3BF8005A2D7">50.1-3 AbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8778256286E50743">50.1-4 ElementaryAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7F43050D8587E767">50.1-5 FreeAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X838DE1AB7B3D70FF">50.1-6 DihedralGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X87865686856910E4">50.1-7 QuaternionGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X86E76B3A796BEFA8">50.1-8 ExtraspecialGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7E54D3E778E6A53E">50.1-9 <span class="Heading">AlternatingGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X858666F97BD85ABB">50.1-10 <span class="Heading">SymmetricGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X788FA7DE84E0FE6A">50.1-11 MathieuGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8469DBBF82F8E5C3">50.1-12 SuzukiGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X87E5B0F679CA7FE4">50.1-13 ReeGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X8674AAA578FE4AEE">50.2 <span class="Heading">Classical Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X85D607DD82AF3E27">50.2-1 <span class="Heading">GeneralLinearGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7CA3F7BF83992C6B">50.2-2 <span class="Heading">SpecialLinearGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X866D4E2B816BDFA5">50.2-3 GeneralUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X82A2AADE805DCDE9">50.2-4 SpecialUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8142A8B07811CA90">50.2-5 <span class="Heading">SymplecticGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7C2051CB7B94CEB1">50.2-6 GeneralOrthogonalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X78D4EEF27AA2DCFD">50.2-7 SpecialOrthogonalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8365E0AB8338DA3F">50.2-8 Omega</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X79C3C61A7D83A6D0">50.2-9 GeneralSemilinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7D3779237CB5B49C">50.2-10 SpecialSemilinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7F0DBEB880D2D574">50.2-11 ProjectiveGeneralLinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X86784EDA80224B74">50.2-12 ProjectiveSpecialLinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7E471ADE7E095604">50.2-13 ProjectiveGeneralUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7A88FE2B7EF9C804">50.2-14 ProjectiveSpecialUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7DEDE2537B8FFFF5">50.2-15 ProjectiveSymplecticGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7F546F907A37DF15">50.2-16 ProjectiveOmega</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X85B9F2D379616C35">50.3 <span class="Heading">Conjugacy Classes in Classical Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X831789117E93171E">50.3-1 NrConjugacyClassesGL</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X817EBD6E841285CD">50.4 <span class="Heading">Constructors for Basic Groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X82676ED5826E9E2E">50.5 <span class="Heading">Selection Functions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X7FBB970F7D110FDB">50.6 <span class="Heading">Transitive Permutation Groups</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7F062EC17EB8287D">50.6-1 TransitiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X871C27427F11B123">50.6-2 NrTransitiveGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7EE614D780C713D1">50.6-3 TransitiveIdentification</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X814D329A7B59F0EB">50.7 <span class="Heading">Small Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8398F2577B719D99">50.7-1 SmallGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7BB133CB7AA8F465">50.7-2 AllSmallGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X875EB1167FF6BA82">50.7-3 OneSmallGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7C587F2A82BEAD19">50.7-4 NumberSmallGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X83044B9D7E3BDF35">50.7-5 IdSmallGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X85352440869327EC">50.7-6 IdsOfAllSmallGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8162304487D0C3E2">50.7-7 IdGap3SolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X833DB8AB80B76D26">50.7-8 SmallGroupsInformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X850CC04E7855FF68">50.7-9 UnloadSmallGroupsData</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X7A884ECF813C2026">50.8 <span class="Heading">Finite Perfect Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X866A25F882A4E97B">50.8-1 SizesPerfectGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7906BBA7818E9415">50.8-2 <span class="Heading">PerfectGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7E1CB2D18085FF9D">50.8-3 PerfectIdentification</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7D68BE547FE5C0F5">50.8-4 NumberPerfectGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7FE695DA86A066E1">50.8-5 NumberPerfectLibraryGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X866356A684F6B15E">50.8-6 SizeNumbersPerfectGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X845419F07BB92867">50.8-7 <span class="Heading">DisplayInformationPerfectGroups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X875C5BE67BAB7F71">50.8-8 <span class="Heading">More about the Perfect Groups Library</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X7AE00EA7791F2574">50.9 <span class="Heading">Primitive Permutation Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7BCEA0C57B6D9F42">50.9-1 PrimitiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8564FECC8477F199">50.9-2 NrPrimitiveGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7B1D4C0483A7F444">50.9-3 PrimitiveGroupsIterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X81329B9B7F5FF8DE">50.9-4 COHORTS_PRIMITIVE_GROUPS</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X7DA239CC848F6CAE">50.10 <span class="Heading">Index numbers of primitive groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X870400597FD4E392">50.10-1 PrimitiveIdentification</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X790D50447ABDF7EE">50.10-2 SimsNo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X784820DA86D0E6F4">50.10-3 PRIMITIVE_INDICES_MAGMA</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X82FD673384BF353B">50.11 <span class="Heading">Irreducible Solvable Matrix Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7DF4B4D683A727E8">50.11-1 IrreducibleSolvableGroupMS</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X836AEF4A7E494724">50.11-2 NumberIrreducibleSolvableGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7DAC64F17C8B49A2">50.11-3 AllIrreducibleSolvableGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X844E60B87FC48D1B">50.11-4 OneIrreducibleSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X81B11EE77EFA745E">50.11-5 PrimitiveIndexIrreducibleSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X816FF4DD8267B4A7">50.11-6 IrreducibleSolvableGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X7873506D873EDB95">50.12 <span class="Heading">Irreducible Maximal Finite Integral Matrix Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8693FD647EF3C53B">50.12-1 ImfNumberQQClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8705F64B7E19DDC7">50.12-2 DisplayImfInvariants</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8604A2167B2E8434">50.12-3 ImfInvariants</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X78935B307B909101">50.12-4 ImfMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X84BF34B27CD5E85C">50.12-5 IsomorphismPermGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7CEDB6CE7BAC4518">50.12-6 IsomorphismPermGroupImfGroup</a></span>
</div></div>
</div>
<h3>50 <span class="Heading">Group Libraries</span></h3>
<p>When you start <strong class="pkg">GAP</strong>, it already knows several groups. Currently <strong class="pkg">GAP</strong> initially knows the following groups:</p>
<ul>
<li><p>some basic groups, such as cyclic groups or symmetric groups (see <a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>),</p>
</li>
<li><p>Classical matrix groups (see <a href="chap50.html#X8674AAA578FE4AEE"><span class="RefLink">50.2</span></a>),</p>
</li>
<li><p>the transitive permutation groups of degree at most 30 (see <a href="chap50.html#X7FBB970F7D110FDB"><span class="RefLink">50.6</span></a>),</p>
</li>
<li><p>a library of groups of small order (see <a href="chap50.html#X814D329A7B59F0EB"><span class="RefLink">50.7</span></a>),</p>
</li>
<li><p>the finite perfect groups of size at most <span class="SimpleMath">10^6</span> (excluding 11 sizes) (see <a href="chap50.html#X7A884ECF813C2026"><span class="RefLink">50.8</span></a>).</p>
</li>
<li><p>the primitive permutation groups of degree <span class="SimpleMath">< 2499</span> (see <a href="chap50.html#X7AE00EA7791F2574"><span class="RefLink">50.9</span></a>),</p>
</li>
<li><p>the irreducible solvable subgroups of <span class="SimpleMath">GL(n,p)</span> for <span class="SimpleMath">n>1</span> and <span class="SimpleMath">p^n < 256</span> (see <a href="chap50.html#X82FD673384BF353B"><span class="RefLink">50.11</span></a>),</p>
</li>
<li><p>the irreducible maximal finite integral matrix groups of dimension at most 31 (see <a href="chap50.html#X7873506D873EDB95"><span class="RefLink">50.12</span></a>),</p>
</li>
<li><p>the crystallographic groups of dimension at most 4</p>
</li>
</ul>
<p>There is usually no relation between the groups in the different libraries and a group may occur in different libraries in different incarnations.</p>
<p>Note that a system administrator may choose to install all, or only a few, or even none of the libraries. So some of the libraries mentioned below may not be available on your installation.</p>
<p><a id="X839981CC7D9B671B" name="X839981CC7D9B671B"></a></p>
<h4>50.1 <span class="Heading">Basic Groups</span></h4>
<p>There are several infinite families of groups which are parametrized by numbers. <strong class="pkg">GAP</strong> provides various functions to construct these groups. The functions always permit (but do not require) one to indicate a filter (see <a href="chap13.html#X84EFA4C07D4277BB"><span class="RefLink">13.2</span></a>), for example <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>) or <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>), in which the group shall be constructed. There always is a default filter corresponding to a "natural" way to describe the group in question. Note that not every group can be constructed in every filter, there may be theoretical restrictions (<code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>) only works for solvable groups) or methods may be available only for a few filters.</p>
<p>Certain filters may admit additional hints. For example, groups constructed in <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>) may be constructed over a specified field, which can be given as second argument of the function that constructs the group; The default field is <code class="func">Rationals</code> (<a href="chap17.html#X7B6029D18570C08A"><span class="RefLink">17.1-1</span></a>).</p>
<p><a id="X8489BECB78664847" name="X8489BECB78664847"></a></p>
<h5>50.1-1 TrivialGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrivialGroup</code>( [<var class="Arg">filter</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a trivial group in the category given by the filter <var class="Arg">filter</var>. If <var class="Arg">filter</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TrivialGroup();</span>
<pc group of size 1 with 0 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">TrivialGroup( IsPermGroup );</span>
Group(())
</pre></div>
<p><a id="X7A7C473D87B31F3B" name="X7A7C473D87B31F3B"></a></p>
<h5>50.1-2 CyclicGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclicGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the cyclic group of size <var class="Arg">n</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclicGroup(12);</span>
<pc group of size 12 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclicGroup(IsPermGroup,12);</span>
Group([ (1,2,3,4,5,6,7,8,9,10,11,12) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">matgrp1:= CyclicGroup( IsMatrixGroup, 12 );</span>
<matrix group of size 12 with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">FieldOfMatrixGroup( matgrp1 );</span>
Rationals
<span class="GAPprompt">gap></span> <span class="GAPinput">matgrp2:= CyclicGroup( IsMatrixGroup, GF(2), 12 );</span>
<matrix group of size 12 with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">FieldOfMatrixGroup( matgrp2 );</span>
GF(2)
</pre></div>
<p><a id="X81CCC3BF8005A2D7" name="X81CCC3BF8005A2D7"></a></p>
<h5>50.1-3 AbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">ints</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs an abelian group in the category given by the filter <var class="Arg">filt</var> which is of isomorphism type <span class="SimpleMath">C_{<var class="Arg">ints</var>[1]} × C_{<var class="Arg">ints</var>[2]} × ... × C_{<var class="Arg">ints</var>[n]}</span>, where <var class="Arg">ints</var> must be a list of positive integers. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). The generators of the group returned are the elements corresponding to the integers in <var class="Arg">ints</var>. For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianGroup([1,2,3]);</span>
<pc group of size 6 with 3 generators>
</pre></div>
<p><a id="X8778256286E50743" name="X8778256286E50743"></a></p>
<h5>50.1-4 ElementaryAbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementaryAbelianGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the elementary abelian group of size <var class="Arg">n</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementaryAbelianGroup(8192);</span>
<pc group of size 8192 with 13 generators>
</pre></div>
<p><a id="X7F43050D8587E767" name="X7F43050D8587E767"></a></p>
<h5>50.1-5 FreeAbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeAbelianGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">rank</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the free abelian group of rank <var class="Arg">n</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsFpGroup</code> (<a href="chap47.html#X850B9DF17D90C3A2"><span class="RefLink">47.1-2</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeAbelianGroup(4);</span>
<fp group on the generators [ f1, f2, f3, f4 ]>
</pre></div>
<p><a id="X838DE1AB7B3D70FF" name="X838DE1AB7B3D70FF"></a></p>
<h5>50.1-6 DihedralGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DihedralGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the dihedral group of size <var class="Arg">n</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DihedralGroup(8);</span>
<pc group of size 8 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">DihedralGroup( IsPermGroup, 8 );</span>
Group([ (1,2,3,4), (2,4) ])
</pre></div>
<p><a id="X87865686856910E4" name="X87865686856910E4"></a></p>
<h5>50.1-7 QuaternionGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuaternionGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DicyclicGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the generalized quaternion group (or dicyclic group) of size <var class="Arg">n</var> in the category given by the filter <var class="Arg">filt</var>. Here, <var class="Arg">n</var> is a multiple of 4. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>). Methods are also available for permutation and matrix groups (of minimal degree and minimal dimension in coprime characteristic).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QuaternionGroup(32);</span>
<pc group of size 32 with 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=QuaternionGroup(IsMatrixGroup,CF(16),32);</span>
Group([ [ [ 0, 1 ], [ -1, 0 ] ], [ [ E(16), 0 ], [ 0, -E(16)^7 ] ] ])
</pre></div>
<p><a id="X86E76B3A796BEFA8" name="X86E76B3A796BEFA8"></a></p>
<h5>50.1-8 ExtraspecialGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExtraspecialGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">order</var>, <var class="Arg">exp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">order</var> be of the form <span class="SimpleMath">p^{2n+1}</span>, for a prime integer <span class="SimpleMath">p</span> and a positive integer <span class="SimpleMath">n</span>. <code class="func">ExtraspecialGroup</code> returns the extraspecial group of order <var class="Arg">order</var> that is determined by <var class="Arg">exp</var>, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <span class="SimpleMath">p</span> is odd then admissible values of <var class="Arg">exp</var> are the exponent of the group (either <span class="SimpleMath">p</span> or <span class="SimpleMath">p^2</span>) or one of <code class="code">'+'</code>, <code class="code">"+"</code>, <code class="code">'-'</code>, <code class="code">"-"</code>. For <span class="SimpleMath">p = 2</span>, only the above plus or minus signs are admissible.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtraspecialGroup( 27, 3 );</span>
<pc group of size 27 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtraspecialGroup( 27, '+' );</span>
<pc group of size 27 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtraspecialGroup( 8, "-" );</span>
<pc group of size 8 with 3 generators>
</pre></div>
<p><a id="X7E54D3E778E6A53E" name="X7E54D3E778E6A53E"></a></p>
<h5>50.1-9 <span class="Heading">AlternatingGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlternatingGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">deg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlternatingGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">dom</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the alternating group of degree <var class="Arg">deg</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>). In the second version, the function constructs the alternating group on the points given in the set <var class="Arg">dom</var> which must be a set of positive integers.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AlternatingGroup(5);</span>
Alt( [ 1 .. 5 ] )
</pre></div>
<p><a id="X858666F97BD85ABB" name="X858666F97BD85ABB"></a></p>
<h5>50.1-10 <span class="Heading">SymmetricGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">deg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">dom</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the symmetric group of degree <var class="Arg">deg</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>). In the second version, the function constructs the symmetric group on the points given in the set <var class="Arg">dom</var> which must be a set of positive integers.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SymmetricGroup(10);</span>
Sym( [ 1 .. 10 ] )
</pre></div>
<p>Note that permutation groups provide special treatment of symmetric and alternating groups, see <a href="chap43.html#X834208CD7C2956A3"><span class="RefLink">43.4</span></a>.</p>
<p><a id="X788FA7DE84E0FE6A" name="X788FA7DE84E0FE6A"></a></p>
<h5>50.1-11 MathieuGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MathieuGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">degree</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the Mathieu group of degree <var class="Arg">degree</var> in the category given by the filter <var class="Arg">filt</var>, where <var class="Arg">degree</var> must be in the set <span class="SimpleMath">{ 9, 10, 11, 12, 21, 22, 23, 24 }</span>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MathieuGroup( 11 );</span>
Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ])
</pre></div>
<p><a id="X8469DBBF82F8E5C3" name="X8469DBBF82F8E5C3"></a></p>
<h5>50.1-12 SuzukiGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SuzukiGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Sz</code>( [<var class="Arg">filt</var>, ]<var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Constructs a group isomorphic to the Suzuki group Sz( <var class="Arg">q</var> ) over the field with <var class="Arg">q</var> elements, where <var class="Arg">q</var> is a non-square power of <span class="SimpleMath">2</span>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the Suzuki group itself. For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SuzukiGroup( 32 );</span>
Sz(32)
</pre></div>
<p><a id="X87E5B0F679CA7FE4" name="X87E5B0F679CA7FE4"></a></p>
<h5>50.1-13 ReeGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReeGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ree</code>( [<var class="Arg">filt</var>, ]<var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Constructs a group isomorphic to the Ree group <span class="SimpleMath">^2G_2(q)</span> where <span class="SimpleMath">q = 3^{1+2m}</span> for <span class="SimpleMath">m</span> a non-negative integer.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>) and the generating matrices are based on <a href="chapBib.html#biBKLM01">[KLM01]</a>. (No particular choice of a generating set is guaranteed.) For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ReeGroup( 27 );</span>
Ree(27)
</pre></div>
<p><a id="X8674AAA578FE4AEE" name="X8674AAA578FE4AEE"></a></p>
<h4>50.2 <span class="Heading">Classical Groups</span></h4>
<p>The following functions return classical groups. For the linear, symplectic, and unitary groups (the latter in dimension at least <span class="SimpleMath">3</span>), the generators are taken from <a href="chapBib.html#biBTay87">[Tay87]</a>. For the unitary groups in dimension <span class="SimpleMath">2</span>, the isomorphism of SU<span class="SimpleMath">(2,q)</span> and SL<span class="SimpleMath">(2,q)</span> is used, see for example <a href="chapBib.html#biBHup67">[Hup67]</a>. The generators of the general and special orthogonal groups are taken from <a href="chapBib.html#biBIshibashiEarnest94">[IE94]</a> and <a href="chapBib.html#biBKleidmanLiebeck90">[KL90]</a>, except that the generators of the groups in odd dimension in even characteristic are constructed via the isomorphism to a symplectic group, see for example <a href="chapBib.html#biBCar72a">[Car72]</a>. The generators of the groups <span class="SimpleMath">Ω^ϵ(d, q)</span> are taken from <a href="chapBib.html#biBRylandsTalor98">[RT98]</a>, except that the generators of SO<span class="SimpleMath">(5, 2)</span> are taken for <span class="SimpleMath">Ω(5, 2)</span>. The generators for the semilinear groups are constructed from the generators of the corresponding linear groups plus one additional generator that describes the action of the group of field automorphisms; for prime integers <span class="SimpleMath">p</span> and positive integers <span class="SimpleMath">f</span>, this yields the matrix groups <span class="SimpleMath">Gamma</span>L<span class="SimpleMath">(d, p^f)</span> and <span class="SimpleMath">Sigma</span>L<span class="SimpleMath">(d, p^f)</span> as groups of <span class="SimpleMath">d f × df</span> matrices over the field with <span class="SimpleMath">p</span> elements.</p>
<p>For symplectic and orthogonal matrix groups returned by the functions described below, the invariant bilinear form is stored as the value of the attribute <code class="func">InvariantBilinearForm</code> (<a href="chap44.html#X7C08385A81AB05E1"><span class="RefLink">44.5-1</span></a>). Analogously, the invariant sesquilinear form defining the unitary groups is stored as the value of the attribute <code class="func">InvariantSesquilinearForm</code> (<a href="chap44.html#X82F22079852130C9"><span class="RefLink">44.5-3</span></a>)). The defining quadratic form of orthogonal groups is stored as the value of the attribute <code class="func">InvariantQuadraticForm</code> (<a href="chap44.html#X7BCACC007EB9B613"><span class="RefLink">44.5-5</span></a>).</p>
<p>Note that due to the different sources for the generators, the invariant forms for the groups <span class="SimpleMath">Ω(e,d,q)</span> are in general different from the forms for SO<span class="SimpleMath">(e,d,q)</span> and GO<span class="SimpleMath">(e,d,q)</span>.</p>
<p><a id="X85D607DD82AF3E27" name="X85D607DD82AF3E27"></a></p>
<h5>50.2-1 <span class="Heading">GeneralLinearGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first two forms construct a group isomorphic to the general linear group GL( <var class="Arg">d</var>, <var class="Arg">R</var> ) of all <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices that are invertible over the ring <var class="Arg">R</var>, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>The third and the fourth form construct the general linear group over the finite field with <var class="Arg">q</var> elements.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the general linear group as a matrix group in its natural action (see also <code class="func">IsNaturalGL</code> (<a href="chap44.html#X86F9A27D7AFAEB5A"><span class="RefLink">44.4-2</span></a>), <code class="func">IsNaturalGLnZ</code> (<a href="chap44.html#X86F9CC1E7DB97CB6"><span class="RefLink">44.6-4</span></a>)).</p>
<p>Currently supported rings <var class="Arg">R</var> are finite fields, the ring <code class="func">Integers</code> (<a href="chap14.html#X853DF11B80068ED5"><span class="RefLink">14</span></a>), and residue class rings <code class="code">Integers mod <var class="Arg">m</var></code>, see <a href="chap14.html#X864BF040862409FC"><span class="RefLink">14.5</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GL(4,3);</span>
GL(4,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">GL(2,Integers);</span>
GL(2,Integers)
<span class="GAPprompt">gap></span> <span class="GAPinput">GL(3,Integers mod 12);</span>
GL(3,Z/12Z)
</pre></div>
<p>Using the <code class="func">OnLines</code> (<a href="chap41.html#X86DC2DD5829CAD9A"><span class="RefLink">41.2-12</span></a>) operation it is possible to obtain the corresponding projective groups in a permutation action:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=GL(4,3);;Size(g);</span>
24261120
<span class="GAPprompt">gap></span> <span class="GAPinput">pgl:=Action(g,Orbit(g,Z(3)^0*[1,0,0,0],OnLines),OnLines);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(pgl);</span>
12130560
</pre></div>
<p>If you are interested only in the projective group as a permutation group and not in the correspondence between its moved points and the points in the projective space, you can also use <code class="func">PGL</code> (<a href="chap50.html#X7F0DBEB880D2D574"><span class="RefLink">50.2-11</span></a>).</p>
<p><a id="X7CA3F7BF83992C6B" name="X7CA3F7BF83992C6B"></a></p>
<h5>50.2-2 <span class="Heading">SpecialLinearGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpecialLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpecialLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first two forms construct a group isomorphic to the special linear group SL( <var class="Arg">d</var>, <var class="Arg">R</var> ) of all those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the ring <var class="Arg">R</var> whose determinant is the identity of <var class="Arg">R</var>, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>The third and the fourth form construct the special linear group over the finite field with <var class="Arg">q</var> elements.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the special linear group as a matrix group in its natural action (see also <code class="func">IsNaturalSL</code> (<a href="chap44.html#X84134F08781EB943"><span class="RefLink">44.4-4</span></a>), <code class="func">IsNaturalSLnZ</code> (<a href="chap44.html#X7B0E70127F5D2EAF"><span class="RefLink">44.6-5</span></a>)).</p>
<p>Currently supported rings <var class="Arg">R</var> are finite fields, the ring <code class="func">Integers</code> (<a href="chap14.html#X853DF11B80068ED5"><span class="RefLink">14</span></a>), and residue class rings <code class="code">Integers mod <var class="Arg">m</var></code>, see <a href="chap14.html#X864BF040862409FC"><span class="RefLink">14.5</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SpecialLinearGroup(2,2);</span>
SL(2,2)
<span class="GAPprompt">gap></span> <span class="GAPinput">SL(3,Integers);</span>
SL(3,Integers)
<span class="GAPprompt">gap></span> <span class="GAPinput">SL(4,Integers mod 4);</span>
SL(4,Z/4Z)
</pre></div>
<p><a id="X866D4E2B816BDFA5" name="X866D4E2B816BDFA5"></a></p>
<h5>50.2-3 GeneralUnitaryGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralUnitaryGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GU</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the general unitary group GU( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <span class="SimpleMath"><var class="Arg">q</var>^2</span> elements that respect a fixed nondegenerate sesquilinear form, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the general unitary group itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneralUnitaryGroup( 3, 5 );</span>
GU(3,5)
</pre></div>
<p><a id="X82A2AADE805DCDE9" name="X82A2AADE805DCDE9"></a></p>
<h5>50.2-4 SpecialUnitaryGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpecialUnitaryGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SU</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the special unitary group GU(<var class="Arg">d</var>, <var class="Arg">q</var>) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <span class="SimpleMath"><var class="Arg">q</var>^2</span> elements whose determinant is the identity of the field and that respect a fixed nondegenerate sesquilinear form, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the special unitary group itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SpecialUnitaryGroup( 3, 5 );</span>
SU(3,5)
</pre></div>
<p><a id="X8142A8B07811CA90" name="X8142A8B07811CA90"></a></p>
<h5>50.2-5 <span class="Heading">SymplecticGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymplecticGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymplecticGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">ring</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Sp</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Sp</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">ring</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SP</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SP</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">ring</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the symplectic group Sp( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements (respectively the ring <var class="Arg">ring</var>) that respect a fixed nondegenerate symplectic form, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the symplectic group itself.</p>
<p>At the moment finite fields or residue class rings <code class="code">Integers mod <var class="Arg">q</var></code>, with <var class="Arg">q</var> an odd prime power are supported.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SymplecticGroup( 4, 2 );</span>
Sp(4,2)
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymplecticGroup(6,Integers mod 9);</span>
Sp(6,Z/9Z)
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(g);</span>
95928796265538862080
</pre></div>
<p><a id="X7C2051CB7B94CEB1" name="X7C2051CB7B94CEB1"></a></p>
<h5>50.2-6 GeneralOrthogonalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralOrthogonalGroup</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GO</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the general orthogonal group GO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements that respect a non-singular quadratic form (see <code class="func">InvariantQuadraticForm</code> (<a href="chap44.html#X7BCACC007EB9B613"><span class="RefLink">44.5-5</span></a>)) specified by <var class="Arg">e</var>, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>The value of <var class="Arg">e</var> must be <span class="SimpleMath">0</span> for odd <var class="Arg">d</var> (and can optionally be omitted in this case), respectively one of <span class="SimpleMath">1</span> or <span class="SimpleMath">-1</span> for even <var class="Arg">d</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the general orthogonal group itself.</p>
<p>Note that in <a href="chapBib.html#biBKleidmanLiebeck90">[KL90]</a>, GO is defined as the stabilizer <span class="SimpleMath">∆(V, F, κ)</span> of the quadratic form, up to scalars, whereas our GO is called <span class="SimpleMath">I(V, F, κ)</span> there.</p>
<p><a id="X78D4EEF27AA2DCFD" name="X78D4EEF27AA2DCFD"></a></p>
<h5>50.2-7 SpecialOrthogonalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpecialOrthogonalGroup</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SO</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SpecialOrthogonalGroup</code> returns a group isomorphic to the special orthogonal group SO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ), which is the subgroup of all those matrices in the general orthogonal group (see <code class="func">GeneralOrthogonalGroup</code> (<a href="chap50.html#X7C2051CB7B94CEB1"><span class="RefLink">50.2-6</span></a>)) that have determinant one, in the category given by the filter <var class="Arg">filt</var>. (The index of SO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) in GO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) is <span class="SimpleMath">2</span> if <var class="Arg">q</var> is odd, and <span class="SimpleMath">1</span> if <var class="Arg">q</var> is even.) Also interesting is the group Omega( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ), see <code class="func">Omega</code> (<a href="chap50.html#X8365E0AB8338DA3F"><span class="RefLink">50.2-8</span></a>), which is always of index <span class="SimpleMath">2</span> in SO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ).</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the special orthogonal group itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneralOrthogonalGroup( 3, 7 );</span>
GO(0,3,7)
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneralOrthogonalGroup( -1, 4, 3 );</span>
GO(-1,4,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">SpecialOrthogonalGroup( 1, 4, 4 );</span>
GO(+1,4,4)
</pre></div>
<p><a id="X8365E0AB8338DA3F" name="X8365E0AB8338DA3F"></a></p>
<h5>50.2-8 Omega</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Omega</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>constructs a group isomorphic to the group <span class="SimpleMath">Ω</span>( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements that respect a non-singular quadratic form (see <code class="func">InvariantQuadraticForm</code> (<a href="chap44.html#X7BCACC007EB9B613"><span class="RefLink">44.5-5</span></a>)) specified by <var class="Arg">e</var>, and that have square spinor norm in odd characteristic or Dickson invariant <span class="SimpleMath">0</span> in even characteristic, respectively, in the category given by the filter <var class="Arg">filt</var>. This group has always index two in SO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ), see <code class="func">SpecialOrthogonalGroup</code> (<a href="chap50.html#X78D4EEF27AA2DCFD"><span class="RefLink">50.2-7</span></a>).</p>
<p>The value of <var class="Arg">e</var> must be <span class="SimpleMath">0</span> for odd <var class="Arg">d</var> (and can optionally be omitted in this case), respectively one of <span class="SimpleMath">1</span> or <span class="SimpleMath">-1</span> for even <var class="Arg">d</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the group <span class="SimpleMath">Ω</span>( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Omega( 3, 5 ); StructureDescription( g );</span>
Omega(0,3,5)
"A5"
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Omega( 1, 4, 4 ); StructureDescription( g );</span>
Omega(+1,4,4)
"A5 x A5"
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Omega( -1, 4, 3 ); StructureDescription( g );</span>
Omega(-1,4,3)
"A6"
</pre></div>
<p><a id="X79C3C61A7D83A6D0" name="X79C3C61A7D83A6D0"></a></p>
<h5>50.2-9 GeneralSemilinearGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralSemilinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GammaL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">GeneralSemilinearGroup</code> returns a group isomorphic to the general semilinear group <span class="SimpleMath">Γ</span>L( <var class="Arg">d</var>, <var class="Arg">q</var> ) of semilinear mappings of the vector space <code class="code">GF( </code><var class="Arg">q</var><code class="code"> )^</code><var class="Arg">d</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group consists of matrices of dimension <var class="Arg">d</var> <span class="SimpleMath">f</span> over the field with <span class="SimpleMath">p</span> elements, where <var class="Arg">q</var> <span class="SimpleMath">= p^f</span>, for a prime integer <span class="SimpleMath">p</span>.</p>
<p><a id="X7D3779237CB5B49C" name="X7D3779237CB5B49C"></a></p>
<h5>50.2-10 SpecialSemilinearGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpecialSemilinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SigmaL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SpecialSemilinearGroup</code> returns a group isomorphic to the special semilinear group <span class="SimpleMath">Σ</span>L( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those semilinear mappings of the vector space <code class="code">GF( </code><var class="Arg">q</var><code class="code"> )^</code><var class="Arg">d</var> (see <code class="func">GeneralSemilinearGroup</code> (<a href="chap50.html#X79C3C61A7D83A6D0"><span class="RefLink">50.2-9</span></a>)) whose linear part has determinant one.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group consists of matrices of dimension <var class="Arg">d</var> <span class="SimpleMath">f</span> over the field with <span class="SimpleMath">p</span> elements, where <var class="Arg">q</var> <span class="SimpleMath">= p^f</span>, for a prime integer <span class="SimpleMath">p</span>.</p>
<p><a id="X7F0DBEB880D2D574" name="X7F0DBEB880D2D574"></a></p>
<h5>50.2-11 ProjectiveGeneralLinearGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveGeneralLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PGL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective general linear group PGL( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements, modulo the centre, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X86784EDA80224B74" name="X86784EDA80224B74"></a></p>
<h5>50.2-12 ProjectiveSpecialLinearGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveSpecialLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PSL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective special linear group PSL( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements whose determinant is the identity of the field, modulo the centre, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X7E471ADE7E095604" name="X7E471ADE7E095604"></a></p>
<h5>50.2-13 ProjectiveGeneralUnitaryGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveGeneralUnitaryGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PGU</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective general unitary group PGU( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <span class="SimpleMath"><var class="Arg">q</var>^2</span> elements that respect a fixed nondegenerate sesquilinear form, modulo the centre, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X7A88FE2B7EF9C804" name="X7A88FE2B7EF9C804"></a></p>
<h5>50.2-14 ProjectiveSpecialUnitaryGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveSpecialUnitaryGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PSU</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective special unitary group PSU( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <span class="SimpleMath"><var class="Arg">q</var>^2</span> elements that respect a fixed nondegenerate sesquilinear form and have determinant 1, modulo the centre, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X7DEDE2537B8FFFF5" name="X7DEDE2537B8FFFF5"></a></p>
<h5>50.2-15 ProjectiveSymplecticGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveSymplecticGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PSP</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PSp</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective symplectic group PSp(<var class="Arg">d</var>,<var class="Arg">q</var>) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements that respect a fixed nondegenerate symplectic form, modulo the centre, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X7F546F907A37DF15" name="X7F546F907A37DF15"></a></p>
<h5>50.2-16 ProjectiveOmega</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveOmega</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ POmega</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective group P<span class="SimpleMath">Ω</span>( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) of <span class="SimpleMath">Ω</span>( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ), modulo the centre (see <code class="func">Omega</code> (<a href="chap50.html#X8365E0AB8338DA3F"><span class="RefLink">50.2-8</span></a>)), in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X85B9F2D379616C35" name="X85B9F2D379616C35"></a></p>
<h4>50.3 <span class="Heading">Conjugacy Classes in Classical Groups</span></h4>
<p>For general and special linear groups (see <code class="func">GeneralLinearGroup</code> (<a href="chap50.html#X85D607DD82AF3E27"><span class="RefLink">50.2-1</span></a>) and <code class="func">SpecialLinearGroup</code> (<a href="chap50.html#X7CA3F7BF83992C6B"><span class="RefLink">50.2-2</span></a>)) <strong class="pkg">GAP</strong> has an efficient method to generate representatives of the conjugacy classes. This uses results from linear algebra on normal forms of matrices. If you know how to do this for other types of classical groups, please, tell us.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := SL(4,9);</span>
SL(4,9)
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClasses(g);</span>
861
<span class="GAPprompt">gap></span> <span class="GAPinput">cl := ConjugacyClasses(g);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(cl);</span>
861
</pre></div>
<p><a id="X831789117E93171E" name="X831789117E93171E"></a></p>
<h5>50.3-1 NrConjugacyClassesGL</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesGL</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesGU</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesSL</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesSU</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesPGL</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesPGU</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesPSL</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesPSU</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesSLIsogeneous</code>( <var class="Arg">n</var>, <var class="Arg">q</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesSUIsogeneous</code>( <var class="Arg">n</var>, <var class="Arg">q</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first of these functions compute for given positive integer <var class="Arg">n</var> and prime power <var class="Arg">q</var> the number of conjugacy classes in the classical groups GL( <var class="Arg">n</var>, <var class="Arg">q</var> ), GU( <var class="Arg">n</var>, <var class="Arg">q</var> ), SL( <var class="Arg">n</var>, <var class="Arg">q</var> ), SU( <var class="Arg">n</var>, <var class="Arg">q</var> ), PGL( <var class="Arg">n</var>, <var class="Arg">q</var> ), PGU( <var class="Arg">n</var>, <var class="Arg">q</var> ), PSL( <var class="Arg">n</var>, <var class="Arg">q</var> ), PSL( <var class="Arg">n</var>, <var class="Arg">q</var> ), respectively. (See also <code class="func">ConjugacyClasses</code> (<a href="chap39.html#X871B570284BBA685"><span class="RefLink">39.10-2</span></a>) and Section <a href="chap50.html#X8674AAA578FE4AEE"><span class="RefLink">50.2</span></a>.)</p>
<p>For each divisor <var class="Arg">f</var> of <var class="Arg">n</var> there is a group of Lie type with the same order as SL( <var class="Arg">n</var>, <var class="Arg">q</var> ), such that its derived subgroup modulo its center is isomorphic to PSL( <var class="Arg">n</var>, <var class="Arg">q</var> ). The various such groups with fixed <var class="Arg">n</var> and <var class="Arg">q</var> are called <em>isogeneous</em>. (Depending on congruence conditions on <var class="Arg">q</var> and <var class="Arg">n</var> several of these groups may actually be isomorphic.) The function <code class="func">NrConjugacyClassesSLIsogeneous</code> computes the number of conjugacy classes in this group. The extreme cases <var class="Arg">f</var> <span class="SimpleMath">= 1</span> and <var class="Arg">f</var> <span class="SimpleMath">= n</span> lead to the groups SL( <var class="Arg">n</var>, <var class="Arg">q</var> ) and PGL( <var class="Arg">n</var>, <var class="Arg">q</var> ), respectively.</p>
<p>The function <code class="func">NrConjugacyClassesSUIsogeneous</code> is the analogous one for the corresponding unitary groups.</p>
<p>The formulae for the number of conjugacy classes are taken from <a href="chapBib.html#biBMac81">[Mac81]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClassesGL(24,27);</span>
22528399544939174406067288580609952
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClassesPSU(19,17);</span>
15052300411163848367708
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClasses(SL(16,16));</span>
1229782938228219920
</pre></div>
<p><a id="X817EBD6E841285CD" name="X817EBD6E841285CD"></a></p>
<h4>50.4 <span class="Heading">Constructors for Basic Groups</span></h4>
<p>All functions described in the previous sections call constructor operations to do the work. The names of the constructors are obtained from the names of the functions by appending <code class="code">"Cons"</code>, so for example <code class="func">CyclicGroup</code> (<a href="chap50.html#X7A7C473D87B31F3B"><span class="RefLink">50.1-2</span></a>) calls the constructor</p>
<p><code class="code">CyclicGroupCons( <var class="Arg">cat</var>, <var class="Arg">n</var> )</code></p>
<p>The first argument <var class="Arg">cat</var> for each method of this constructor must be the category for which the method is installed. For example the method for constructing a cyclic permutation group is installed as follows (see <code class="func">InstallMethod</code> (<a href="chap78.html#X837EFDAB7BEF290B"><span class="RefLink">78.2-1</span></a>) for the meaning of the arguments.</p>
<div class="example"><pre>
InstallMethod( CyclicGroupCons,
"regular perm group",
true,
[ IsPermGroup and IsRegularProp and IsFinite, IsInt and IsPosRat ], 0,
function( filter, n )
...
end );
</pre></div>
<p><a id="X82676ED5826E9E2E" name="X82676ED5826E9E2E"></a></p>
<h4>50.5 <span class="Heading">Selection Functions</span></h4>
<p><code class="code">All<var class="Arg">Library</var>Groups( <var class="Arg">fun1</var>, <var class="Arg">val1</var>, ... )</code></p>
<p>For a number of the group libraries two <em>selection functions</em> are provided. Each <code class="code">All<var class="Arg">Library</var>Groups</code> selection function permits one to select <em>all</em> groups from the library <var class="Arg">Library</var> that have a given set of properties. Currently, the library selection functions provided, of this type, are <code class="func">AllSmallGroups</code> (<a href="chap50.html#X7BB133CB7AA8F465"><span class="RefLink">50.7-2</span></a>), <code class="func">AllIrreducibleSolvableGroups</code> (<a href="chap50.html#X7DAC64F17C8B49A2"><span class="RefLink">50.11-3</span></a>), <code class="code">AllTransitiveGroups</code>, and <code class="code">AllPrimitiveGroups</code>. Corresponding to each of these there is a <code class="code">One<var class="Arg">Library</var>Group</code> function (see below) which returns at most one group.</p>
<p>These functions take an arbitrary number of pairs (but at least one pair) of arguments. The first argument in such a pair is a function that can be applied to the groups in the library, and the second argument is either a single value that this function must return in order to have this group included in the selection, or a list of such values. For the function <code class="func">AllSmallGroups</code> (<a href="chap50.html#X7BB133CB7AA8F465"><span class="RefLink">50.7-2</span></a>) the first such function must be <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>), and, unlike the other library selection functions, it supports an alternative syntax where <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>) is omitted (see <code class="func">AllSmallGroups</code> (<a href="chap50.html#X7BB133CB7AA8F465"><span class="RefLink">50.7-2</span></a>)). Also, see <code class="func">AllIrreducibleSolvableGroups</code> (<a href="chap50.html#X7DAC64F17C8B49A2"><span class="RefLink">50.11-3</span></a>), for details pertaining to this function.</p>
<p>For an example, let us consider the selection function for the library of transitive groups (also see <a href="chap50.html#X7FBB970F7D110FDB"><span class="RefLink">50.6</span></a>). The command</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllTransitiveGroups(NrMovedPoints,[10..15],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Size, [1..100],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> IsAbelian, false );</span>
</pre></div>
<p>returns a list of all transitive groups with degree between 10 and 15 and size less than 100 that are not abelian.</p>
<p>Thus <code class="code">AllTransitiveGroups</code> behaves as if it was implemented by a function similar to the one defined below, where <code class="code">TransitiveGroupsList</code> is a list of all transitive groups. (Note that in the definition below we assume for simplicity that <code class="code">AllTransitiveGroups</code> accepts exactly 4 arguments. It is of course obvious how to change this definition so that the function would accept a variable number of arguments.)</p>
<div class="example"><pre>
AllTransitiveGroups := function( fun1, val1, fun2, val2 )
local groups, g, i;
groups := [];
for i in [ 1 .. Length( TransitiveGroupsList ) ] do
g := TransitiveGroupsList[i];
if fun1(g) = val1 or IsList(val1) and fun1(g) in val1
and fun2(g) = val2 or IsList(val2) and fun2(g) in val2
then
Add( groups, g );
fi;
od;
return groups;
end;
</pre></div>
<p>Note that the real selection functions are considerably more difficult, to improve the efficiency. Most important, each recognizes a certain set of properties which are precomputed for the library without having to compute them anew for each group. This will substantially speed up the selection process. In the description of each library we will list the properties that are stored for this library.</p>
<p><code class="code">One<var class="Arg">Library</var>Group( <var class="Arg">fun1</var>, <var class="Arg">val1</var>, ... )</code></p>
<p>For each <code class="code">All<var class="Arg">Library</var>Groups</code> function (see above) there is a corresponding function <code class="code">One<var class="Arg">Library</var>Group</code> on exactly the same arguments, i.e., there are <code class="code">OneSmallGroup</code>, <code class="code">OneIrreducibleSolvableGroup</code>, <code class="code">OneTransitiveGroup</code>, and <code class="code">OnePrimitiveGroup</code>. Each function simply returns <em>one</em> group in the library that has the prescribed properties, instead of <em>all</em> such groups. It returns <code class="keyw">fail</code> if no such group exists in the library.</p>
<p><a id="X7FBB970F7D110FDB" name="X7FBB970F7D110FDB"></a></p>
<h4>50.6 <span class="Heading">Transitive Permutation Groups</span></h4>
<p>The transitive groups library currently contains representatives for all transitive permutation groups of degree at most 30. Two permutations groups of the same degree are considered to be equivalent, if there is a renumbering of points, which maps one group into the other one. In other words, if they lie in the same conjugacy class under operation of the full symmetric group by conjugation.</p>
<p>The selection functions (see <a href="chap50.html#X82676ED5826E9E2E"><span class="RefLink">50.5</span></a>) for the transitive groups library are <code class="code">AllTransitiveGroups</code> and <code class="code">OneTransitiveGroup</code>. They obtain the following attributes from the database without having to compute them anew:</p>
<p><code class="func">NrMovedPoints</code> (<a href="chap42.html#X85E7B1E28430F49E"><span class="RefLink">42.3-4</span></a>), <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>), <code class="func">Transitivity</code> (<a href="chap41.html#X8295D733796B7A37"><span class="RefLink">41.10-2</span></a>), and <code class="func">IsPrimitive</code> (<a href="chap41.html#X84C19AD68247B760"><span class="RefLink">41.10-7</span></a>).</p>
<p>This library was computed by Gregory Butler, John McKay, Gordon Royle and Alexander Hulpke. The list of transitive groups up to degree 11 was published in <a href="chapBib.html#biBBM83">[BM83]</a>, the list of degree 12 was published in <a href="chapBib.html#biBRoy87">[Roy87]</a>, degree 14 and 15 were published in <a href="chapBib.html#biBButler93">[But93]</a> and degrees 16-30 were published in <a href="chapBib.html#biBHulpke96">[Hul96]</a> and <a href="chapBib.html#biBHulpkeTG">[Hul05]</a>. (Groups of prime degree of course are primitive and were known long before.)</p>
<p>The arrangement and the names of the groups of degree up to 15 is the same as given in <a href="chapBib.html#biBConwayHulpkeMcKay98">[CHM98]</a>. With the exception of the symmetric and alternating group (which are represented as <code class="func">SymmetricGroup</code> (<a href="chap50.html#X858666F97BD85ABB"><span class="RefLink">50.1-10</span></a>) and <code class="func">AlternatingGroup</code> (<a href="chap50.html#X7E54D3E778E6A53E"><span class="RefLink">50.1-9</span></a>)) the generators for these groups also conform to this paper with the only difference that 0 (which is not permitted in <strong class="pkg">GAP</strong> for permutations to act on) is always replaced by the degree.</p>
<p><a id="X7F062EC17EB8287D" name="X7F062EC17EB8287D"></a></p>
<h5>50.6-1 TransitiveGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransitiveGroup</code>( <var class="Arg">deg</var>, <var class="Arg">nr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the <var class="Arg">nr</var>-th transitive group of degree <var class="Arg">deg</var>. Both <var class="Arg">deg</var> and <var class="Arg">nr</var> must be positive integers. The transitive groups of equal degree are sorted with respect to their size, so for example <code class="code">TransitiveGroup( <var class="Arg">deg</var>, 1 )</code> is a transitive group of degree and size <var class="Arg">deg</var>, e.g, the cyclic group of size <var class="Arg">deg</var>, if <var class="Arg">deg</var> is a prime.</p>
<p><a id="X871C27427F11B123" name="X871C27427F11B123"></a></p>
<h5>50.6-2 NrTransitiveGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrTransitiveGroups</code>( <var class="Arg">deg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the number of transitive groups of degree <var class="Arg">deg</var> stored in the library of transitive groups. The function returns <code class="keyw">fail</code> if <var class="Arg">deg</var> is beyond the range of the library.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TransitiveGroup(10,22);</span>
S(5)[x]2
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=AllTransitiveGroups(NrMovedPoints,12,Size,1440,IsSolvable,false);</span>
[ S(6)[x]2, M_10.2(12)=A_6.E_4(12)=[S_6[1/720]{M_10}S_6]2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(l,IsSolvable);</span>
[ false, false ]
</pre></div>
<p><a id="X7EE614D780C713D1" name="X7EE614D780C713D1"></a></p>
<h5>50.6-3 TransitiveIdentification</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransitiveIdentification</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a permutation group, acting transitively on a set of up to 30 points. Then <code class="func">TransitiveIdentification</code> will return the position of this group in the transitive groups library. This means, if <var class="Arg">G</var> acts on <span class="SimpleMath">m</span> points and <code class="func">TransitiveIdentification</code> returns <span class="SimpleMath">n</span>, then <var class="Arg">G</var> is permutation isomorphic to the group <code class="code">TransitiveGroup(m,n)</code>.</p>
<p>Note: The points moved do <em>not</em> need to be [1..<var class="Arg">n</var>], the group <span class="SimpleMath">⟨ (2,3,4),(2,3) ⟩</span> is considered to be transitive on 3 points. If the group has several orbits on the points moved by it the result of <code class="func">TransitiveIdentification</code> is undefined.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TransitiveIdentification(Group((1,2),(1,2,3)));</span>
2
</pre></div>
<p><a id="X814D329A7B59F0EB" name="X814D329A7B59F0EB"></a></p>
<h4>50.7 <span class="Heading">Small Groups</span></h4>
<p>The Small Groups library gives access to all groups of certain "small" orders. The groups are sorted by their orders and they are listed up to isomorphism; that is, for each of the available orders a complete and irredundant list of isomorphism type representatives of groups is given. Currently, the library contains the following groups:</p>
<ul>
<li><p>those of order at most 2000 except 1024 (<span class="SimpleMath">423 164 062</span> groups);</p>
</li>
<li><p>those of cubefree order at most 50 000 (<span class="SimpleMath">395 703</span> groups);</p>
</li>
<li><p>those of order <span class="SimpleMath">p^7</span> for the primes <span class="SimpleMath">p = 3,5,7,11</span> (<span class="SimpleMath">907 489</span> groups);</p>
</li>
<li><p>those of order <span class="SimpleMath">p^n</span> for <span class="SimpleMath">n ≤ 6</span> and all primes <span class="SimpleMath">p</span></p>
</li>
<li><p>those of order <span class="SimpleMath">q^n ⋅ p</span> for <span class="SimpleMath">q^n</span> dividing <span class="SimpleMath">2^8</span>, <span class="SimpleMath">3^6</span>, <span class="SimpleMath">5^5</span> or <span class="SimpleMath">7^4</span> and all primes <span class="SimpleMath">p</span> with <span class="SimpleMath">p ≠ q</span>;</p>
</li>
<li><p>those of squarefree order;</p>
</li>
<li><p>those whose order factorises into at most 3 primes.</p>
</li>
</ul>
<p>The first three items in this list cover an explicit range of orders; the last four provide access to infinite families of groups having orders of certain types.</p>
<p>The library also has an identification function: it returns the library number of a given group. This function determines library numbers using invariants of groups. The function is available for all orders in the library except for the orders 512 and 1536 and except for the orders <span class="SimpleMath">p^5</span>, <span class="SimpleMath">p^6</span> and <span class="SimpleMath">p^7</span> above 2000.</p>
<p>The library is organised in 11 layers. Each layer contains the groups of certain orders and their corresponding group identification routines. It is possible to install the first <span class="SimpleMath">n</span> layers of the group library and the first <span class="SimpleMath">m</span> layers of the group identification for each <span class="SimpleMath">1 ≤ m ≤ n ≤ 11</span>. This might be useful to save disk space. There is an extensive <code class="file">README</code> file for the Small Groups library available in the <code class="code">small</code> directory of the <strong class="pkg">GAP</strong> distribution containing detailed information on the layers. A brief description of the layers is given here:</p>
<dl>
<dt><strong class="Mark">(1)</strong></dt>
<dd><p>the groups whose order factorises into at most 3 primes.</p>
</dd>
<dt><strong class="Mark">(2)</strong></dt>
<dd><p>the remaining groups of order at most 1000 except 512 and 768.</p>
</dd>
<dt><strong class="Mark">(3)</strong></dt>
<dd><p>the remaining groups of order <span class="SimpleMath">2^n ⋅ p</span> with <span class="SimpleMath">n ≤ 8</span> and <span class="SimpleMath">p</span> an odd prime.</p>
</dd>
<dt><strong class="Mark">(4)</strong></dt>
<dd><p>the remaining groups of order <span class="SimpleMath">5^5</span>, <span class="SimpleMath">7^4</span> and of order <span class="SimpleMath">q^n ⋅ p</span> for <span class="SimpleMath">q^n</span> dividing <span class="SimpleMath">3^6</span>, <span class="SimpleMath">5^5</span> or <span class="SimpleMath">7^4</span> and <span class="SimpleMath">p ≠ q</span> a prime.</p>
</dd>
<dt><strong class="Mark">(5)</strong></dt>
<dd><p>the remaining groups of order at most 2000 except 1024, 1152, 1536 and 1920.</p>
</dd>
<dt><strong class="Mark">(6)</strong></dt>
<dd><p>the groups of orders 1152 and 1920.</p>
</dd>
<dt><strong class="Mark">(7)</strong></dt>
<dd><p>the groups of order 512.</p>
</dd>
<dt><strong class="Mark">(8)</strong></dt>
<dd><p>the groups of order 1536.</p>
</dd>
<dt><strong class="Mark">(9)</strong></dt>
<dd><p>the remaining groups of order <span class="SimpleMath">p^n</span> for <span class="SimpleMath">4 ≤ n ≤ 6</span>.</p>
</dd>
<dt><strong class="Mark">(10)</strong></dt>
<dd><p>the remaining groups of cubefree order at most 50 000 and of squarefree order.</p>
</dd>
<dt><strong class="Mark">(11)</strong></dt>
<dd><p>the remaining groups of order <span class="SimpleMath">p^7</span> for <span class="SimpleMath">p = 3,5,7,11</span>.</p>
</dd>
</dl>
<p>The data in this library has been carefully checked and cross-checked. It is believed to be reliable. However, no absolute guarantees are given and users should, as always, make their own checks in critical cases.</p>
<p>The data occupies about 30 MB (storing over 400 million groups in about 200 megabits). The group identification occupies about 47 MB of which 18 MB is used for the groups in layer (6). More information on the Small Groups library can be found on <span class="URL"><a href="http://www.icm.tu-bs.de/ag_algebra/software/small/">http://www.icm.tu-bs.de/ag_algebra/software/small/</a></span></p>
<p>This library has been constructed by Hans Ulrich Besche, Bettina Eick and E. A. O'Brien. A survey on this topic and an account of the history of group constructions can be found in <a href="chapBib.html#biBBEO01">[BEO02]</a>. Further detailed information on the construction of this library is available in <a href="chapBib.html#biBNew77">[New77]</a>, <a href="chapBib.html#biBOBr90">[O'B90]</a>, <a href="chapBib.html#biBOBr91">[O'B91]</a>, <a href="chapBib.html#biBBescheEick98">[BE99a]</a>, <a href="chapBib.html#biBBescheEick1000">[BE99b]</a>, <a href="chapBib.html#biBBescheEick768">[BE01]</a>, <a href="chapBib.html#biBBEO00">[BEO01]</a>, <a href="chapBib.html#biBEOB99">[EO99a]</a>, <a href="chapBib.html#biBEOB98">[EO99b]</a>, <a href="chapBib.html#biBNOV04">[NOV04]</a>, <a href="chapBib.html#biBGir03">[Gir03]</a>, <a href="chapBib.html#biBDEi05">[DE05]</a>, <a href="chapBib.html#biBOV05">[OV05]</a>. The Small Groups library incorporates the <strong class="pkg">GAP</strong> 3 libraries <code class="code">TwoGroup</code> and <code class="code">ThreeGroup</code>. The data from these libraries was directly included into the Small Groups library, and the ordering there was preserved. The Small Groups library replaces the Gap 3 library of solvable groups of order at most 100. However, both the organisation and data descriptions of these groups has changed in the Small Groups library.</p>
<p><a id="X8398F2577B719D99" name="X8398F2577B719D99"></a></p>
<h5>50.7-1 SmallGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallGroup</code>( <var class="Arg">order</var>, <var class="Arg">i</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallGroup</code>( <var class="Arg">pair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the <var class="Arg">i</var>-th group of order <var class="Arg">order</var> in the catalogue. If the group is solvable, it will be given as a PcGroup; otherwise it will be given as a permutation group. If the groups of order <var class="Arg">order</var> are not installed, the function reports an error and enters a break loop.</p>
<p><a id="X7BB133CB7AA8F465" name="X7BB133CB7AA8F465"></a></p>
<h5>50.7-2 AllSmallGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSmallGroups</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns all groups with certain properties as specified by <var class="Arg">arg</var>. If <var class="Arg">arg</var> is a number <span class="SimpleMath">n</span>, then this function returns all groups of order <span class="SimpleMath">n</span>. However, the function can also take several arguments which then must be organized in pairs <code class="code">function</code> and <code class="code">value</code>. In this case the first function must be <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>) and the first value an order or a range of orders. If value is a list then it is considered a list of possible function values to include. The function returns those groups of the specified orders having those properties specified by the remaining functions and their values.</p>
<p>Precomputed information is stored for the properties <code class="func">IsAbelian</code> (<a href="chap35.html#X830A4A4C795FBC2D"><span class="RefLink">35.4-9</span></a>), <code class="func">IsNilpotentGroup</code> (<a href="chap39.html#X87D062608719F2CD"><span class="RefLink">39.15-3</span></a>), <code class="func">IsSupersolvableGroup</code> (<a href="chap39.html#X7AADF2E88501B9FF"><span class="RefLink">39.15-8</span></a>), <code class="func">IsSolvableGroup</code> (<a href="chap39.html#X809C78D5877D31DF"><span class="RefLink">39.15-6</span></a>), <code class="func">RankPGroup</code> (<a href="chap39.html#X840A4F937ABF15E1"><span class="RefLink">39.15-22</span></a>), <code class="func">PClassPGroup</code> (<a href="chap39.html#X863434AD7DDE514B"><span class="RefLink">39.15-21</span></a>), <code class="func">LGLength</code> (<a href="chap45.html#X7C3912F77B12C8B6"><span class="RefLink">45.13-6</span></a>), <code class="code">FrattinifactorSize</code> and <code class="code">FrattinifactorId</code> for the groups of order at most <span class="SimpleMath">2000</span> which have more than three prime factors, except those of order <span class="SimpleMath">512</span>, <span class="SimpleMath">768</span>, <span class="SimpleMath">1024</span>, <span class="SimpleMath">1152</span>, <span class="SimpleMath">1536</span>, <span class="SimpleMath">1920</span> and those of order <span class="SimpleMath">p^n ⋅ q > 1000</span> with <span class="SimpleMath">n > 2</span>.</p>
<p><a id="X875EB1167FF6BA82" name="X875EB1167FF6BA82"></a></p>
<h5>50.7-3 OneSmallGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneSmallGroup</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns one group with certain properties as specified by <var class="Arg">arg</var>. The permitted arguments are those supported by <code class="func">AllSmallGroups</code> (<a href="chap50.html#X7BB133CB7AA8F465"><span class="RefLink">50.7-2</span></a>).</p>
<p><a id="X7C587F2A82BEAD19" name="X7C587F2A82BEAD19"></a></p>
<h5>50.7-4 NumberSmallGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumberSmallGroups</code>( <var class="Arg">order</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the number of groups of order <var class="Arg">order</var>.</p>
<p><a id="X83044B9D7E3BDF35" name="X83044B9D7E3BDF35"></a></p>
<h5>50.7-5 IdSmallGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdSmallGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the library number of <var class="Arg">G</var>; that is, the function returns a pair <code class="code">[<var class="Arg">order</var>, <var class="Arg">i</var>]</code> where <var class="Arg">G</var> is isomorphic to <code class="code">SmallGroup( <var class="Arg">order</var>, <var class="Arg">i</var> )</code>.</p>
<p><a id="X85352440869327EC" name="X85352440869327EC"></a></p>
<h5>50.7-6 IdsOfAllSmallGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdsOfAllSmallGroups</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>similar to <code class="code">AllSmallGroups</code> but returns ids instead of groups. This may prevent workspace overflows, if a large number of groups are expected in the output.</p>
<p><a id="X8162304487D0C3E2" name="X8162304487D0C3E2"></a></p>
<h5>50.7-7 IdGap3SolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdGap3SolvableGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gap3CatalogueIdGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the catalogue number of <var class="Arg">G</var> in the <strong class="pkg">GAP</strong> 3 catalogue of solvable groups; that is, the function returns a pair <code class="code">[<var class="Arg">order</var>, <var class="Arg">i</var>]</code> meaning that <var class="Arg">G</var> is isomorphic to the group <code class="code">SolvableGroup( <var class="Arg">order</var>, <var class="Arg">i</var> )</code> in <strong class="pkg">GAP</strong> 3.</p>
<p><a id="X833DB8AB80B76D26" name="X833DB8AB80B76D26"></a></p>
<h5>50.7-8 SmallGroupsInformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallGroupsInformation</code>( <var class="Arg">order</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>prints information on the groups of the specified order.</p>
<p><a id="X850CC04E7855FF68" name="X850CC04E7855FF68"></a></p>
<h5>50.7-9 UnloadSmallGroupsData</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnloadSmallGroupsData</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p><strong class="pkg">GAP</strong> loads all necessary data from the library automatically, but it does not delete the data from the workspace again. Usually, this will be not necessary, since the data is stored in a compressed format. However, if a large number of groups from the library have been loaded, then the user might wish to remove the data from the workspace and this can be done by the above function call.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SmallGroup( 768, 1000000 );</span>
<pc group of size 768 with 9 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SmallGroup( [768, 1000000] );</span>
<pc group of size 768 with 9 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllSmallGroups( 6 );</span>
[ <pc group of size 6 with 2 generators>,
<pc group of size 6 with 2 generators> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AllSmallGroups( Size, 120, IsSolvableGroup, false );</span>
[ Group(
[ (1,2,4,8)(3,6,9,5)(7,12,13,17)(10,14,11,15)(16,20,21,24)(18,22,
19,23), (1,3,7)(2,5,10)(4,9,13)(6,11,8)(12,16,20)(14,18,
22)(15,19,23)(17,21,24) ]), Group([ (1,2,3,4,5), (1,2) ]),
Group([ (1,2,3,5,4), (1,3)(2,4)(6,7) ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G := OneSmallGroup( 120, IsNilpotentGroup, false );</span>
<pc group of size 120 with 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallGroup(G);</span>
[ 120, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G := OneSmallGroup( Size, [1..1000], IsSolvableGroup, false );</span>
Group([ (1,2,3,4,5), (1,2,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallGroup(G);</span>
[ 60, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">UnloadSmallGroupsData();</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallGroup( GL( 2,3 ) );</span>
[ 48, 29 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallGroup( Group( (1,2,3,4),(4,5) ) );</span>
[ 120, 34 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdsOfAllSmallGroups( Size, 60, IsSupersolvableGroup, true );</span>
[ [ 60, 1 ], [ 60, 2 ], [ 60, 3 ], [ 60, 4 ], [ 60, 6 ], [ 60, 7 ],
[ 60, 8 ], [ 60, 10 ], [ 60, 11 ], [ 60, 12 ], [ 60, 13 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberSmallGroups( 512 );</span>
10494213
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberSmallGroups( 2^8 * 23 );</span>
1083472
</pre></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberSmallGroups( 2^9 * 23 );</span>
Error, the library of groups of size 11776 is not available called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
<span class="GAPbrkprompt">brk></span> <span class="GAPinput">quit;</span>
gap>
</pre></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallGroupsInformation( 32 );</span>
There are 51 groups of order 32.
They are sorted by their ranks.
1 is cyclic.
2 - 20 have rank 2.
21 - 44 have rank 3.
45 - 50 have rank 4.
51 is elementary abelian.
For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and
FrattinifactorId.
This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.
</pre></div>
<p><a id="X7A884ECF813C2026" name="X7A884ECF813C2026"></a></p>
<h4>50.8 <span class="Heading">Finite Perfect Groups</span></h4>
<p>The <strong class="pkg">GAP</strong> library of finite perfect groups provides, up to isomorphism, a list of all perfect groups whose sizes are less than <span class="SimpleMath">10^6</span> excluding the following sizes:</p>
<ul>
<li><p>For <span class="SimpleMath">n = 61440</span>, 122880, 172032, 245760, 344064, 491520, 688128, or 983040, the perfect groups of size <span class="SimpleMath">n</span> have not completely been determined yet. The library neither provides the number of these groups nor the groups themselves.</p>
</li>
<li><p>For <span class="SimpleMath">n = 86016</span>, 368640, or 737280, the library does not yet contain the perfect groups of size <span class="SimpleMath">n</span>, it only provides their numbers which are 52, 46, and 54, respectively.</p>
</li>
</ul>
<p>Except for these eleven sizes, the list of altogether 1097 perfect groups in the library is complete. It relies on results of Derek F. Holt and Wilhelm Plesken which are published in their book "Perfect Groups" <a href="chapBib.html#biBHP89">[HP89]</a>. Moreover, they have supplied us with files with presentations of 488 of the groups. In terms of these, the remaining 607 nontrivial groups in the library can be described as 276 direct products, 107 central products, and 224 subdirect products. They are computed automatically by suitable <strong class="pkg">GAP</strong> functions whenever they are needed. Two additional groups omitted from the book "Perfect Groups" have also been included.</p>
<p>We are grateful to Derek Holt and Wilhelm Plesken for making their groups available to the <strong class="pkg">GAP</strong> community by contributing their files. It should be noted that their book contains a lot of further information for many of the library groups. So we would like to recommend it to any <strong class="pkg">GAP</strong> user who is interested in the groups.</p>
<p>The library has been brought into <strong class="pkg">GAP</strong> format by Volkmar Felsch.</p>
<p>As all groups are stored by presentations, a permutation representation is obtained by coset enumeration. Note that some of the library groups do not have a faithful permutation representation of small degree. Computations in these groups may be rather time consuming.</p>
<p><a id="X866A25F882A4E97B" name="X866A25F882A4E97B"></a></p>
<h5>50.8-1 SizesPerfectGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SizesPerfectGroups</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the ordered list of all numbers up to <span class="SimpleMath">10^6</span> that occur as sizes of perfect groups. One can iterate over the perfect groups library with:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">for n in SizesPerfectGroups() do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> for k in [1..NrPerfectLibraryGroups(n)] do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> pg := PerfectGroup(n,k);</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
</pre></div>
<p><a id="X7906BBA7818E9415" name="X7906BBA7818E9415"></a></p>
<h5>50.8-2 <span class="Heading">PerfectGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PerfectGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">size</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PerfectGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">sizenumberpair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a group which is isomorphic to the library group specified by the size number <code class="code">[ <var class="Arg">size</var>, <var class="Arg">n</var> ]</code> or by the two separate arguments <var class="Arg">size</var> and <var class="Arg">n</var>, assuming a default value of <span class="SimpleMath"><var class="Arg">n</var> = 1</span>. The optional argument <var class="Arg">filt</var> defines the filter in which the group is returned. Possible filters so far are <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>) and <code class="func">IsSubgroupFpGroup</code> (<a href="chap47.html#X7AF7E2B48199452C"><span class="RefLink">47.1-1</span></a>). In the latter case, the generators and relators used coincide with those given in <a href="chapBib.html#biBHP89">[HP89]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := PerfectGroup(IsPermGroup,6048,1);</span>
U3(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=PerfectGroup(IsPermGroup,823080,2);</span>
A5 2^1 19^2 C 19^1
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints(G);</span>
6859
</pre></div>
<p><a id="X7E1CB2D18085FF9D" name="X7E1CB2D18085FF9D"></a></p>
<h5>50.8-3 PerfectIdentification</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PerfectIdentification</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute is set for all groups obtained from the perfect groups library and has the value <code class="code">[<var class="Arg">size</var>,<var class="Arg">nr</var>]</code> if the group is obtained with these parameters from the library.</p>
<p><a id="X7D68BE547FE5C0F5" name="X7D68BE547FE5C0F5"></a></p>
<h5>50.8-4 NumberPerfectGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumberPerfectGroups</code>( <var class="Arg">size</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the number of non-isomorphic perfect groups of size <var class="Arg">size</var> for each positive integer <var class="Arg">size</var> up to <span class="SimpleMath">10^6</span> except for the eight sizes listed at the beginning of this section for which the number is not yet known. For these values as well as for any argument out of range it returns <code class="keyw">fail</code>.</p>
<p><a id="X7FE695DA86A066E1" name="X7FE695DA86A066E1"></a></p>
<h5>50.8-5 NumberPerfectLibraryGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumberPerfectLibraryGroups</code>( <var class="Arg">size</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the number of perfect groups of size <var class="Arg">size</var> which are available in the library of finite perfect groups. (The purpose of the function is to provide a simple way to formulate a loop over all library groups of a given size.)</p>
<p><a id="X866356A684F6B15E" name="X866356A684F6B15E"></a></p>
<h5>50.8-6 SizeNumbersPerfectGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SizeNumbersPerfectGroups</code>( <var class="Arg">factor1</var>, <var class="Arg">factor2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SizeNumbersPerfectGroups</code> returns a list of pairs, each entry consisting of a group order and the number of those groups in the library of perfect groups that contain the specified factors <var class="Arg">factor1</var>, <var class="Arg">factor2</var>, ... among their composition factors.</p>
<p>Each argument must either be the name of a simple group or an integer which stands for the product of the sizes of one or more cyclic factors. (In fact, the function replaces all integers among the arguments by their product.)</p>
<p>The following text strings are accepted as simple group names.</p>
<ul>
<li><p><code class="code">A<var class="Arg">n</var></code> or <code class="code">A(<var class="Arg">n</var>)</code> for the alternating groups <span class="SimpleMath">A_<var class="Arg">n</var></span>, <span class="SimpleMath">5 ≤ n ≤ 9</span>, for example <code class="code">A5</code> or <code class="code">A(6)</code>.</p>
</li>
<li><p><code class="code">L<var class="Arg">n</var>(<var class="Arg">q</var>)</code> or <code class="code">L(<var class="Arg">n</var>,<var class="Arg">q</var>)</code> for PSL<span class="SimpleMath">(n,q)</span>, where <span class="SimpleMath">n ∈ { 2, 3 }</span> and <span class="SimpleMath">q</span> a prime power, ranging</p>
<ul>
<li><p>for <span class="SimpleMath">n = 2</span> from 4 to 125</p>
</li>
<li><p>for <span class="SimpleMath">n = 3</span> from 2 to 5</p>
</li>
</ul>
</li>
<li><p><code class="code">U<var class="Arg">n</var>(<var class="Arg">q</var>)</code> or <code class="code">U(<var class="Arg">n</var>,<var class="Arg">q</var>)</code> for PSU<span class="SimpleMath">(n,q)</span>, where <span class="SimpleMath">n ∈ { 3, 4 }</span> and <span class="SimpleMath">q</span> a prime power, ranging</p>
<ul>
<li><p>for <span class="SimpleMath">n = 3</span> from 3 to 5</p>
</li>
<li><p>for <span class="SimpleMath">n = 4</span> from 2 to 2</p>
</li>
</ul>
</li>
<li><p><code class="code">Sp4(4)</code> or <code class="code">S(4,4)</code> for the symplectic group Sp<span class="SimpleMath">(4,4)</span>,</p>
</li>
<li><p><code class="code">Sz(8)</code> for the Suzuki group Sz<span class="SimpleMath">(8)</span>,</p>
</li>
<li><p><code class="code">M<var class="Arg">n</var></code> or <code class="code">M(<var class="Arg">n</var>)</code> for the Mathieu groups <span class="SimpleMath">M_11</span>, <span class="SimpleMath">M_12</span>, and <span class="SimpleMath">M_22</span>, and</p>
</li>
<li><p><code class="code">J<var class="Arg">n</var></code> or <code class="code">J(<var class="Arg">n</var>)</code> for the Janko groups <span class="SimpleMath">J_1</span> and <span class="SimpleMath">J_2</span>.</p>
</li>
</ul>
<p>Note that, for most of the groups, the preceding list offers two different names in order to be consistent with the notation used in <a href="chapBib.html#biBHP89">[HP89]</a> as well as with the notation used in the <code class="func">DisplayCompositionSeries</code> (<a href="chap39.html#X82C0D0217ACB2042"><span class="RefLink">39.17-6</span></a>) command of <strong class="pkg">GAP</strong>. However, as the names are compared as text strings, you are restricted to the above choice. Even expressions like <code class="code">L2(2^5)</code> are not accepted.</p>
<p>As the use of the term PSU<span class="SimpleMath">(n,q)</span> is not unique in the literature, we mention that in this library it denotes the factor group of SU<span class="SimpleMath">(n,q)</span> by its centre, where SU<span class="SimpleMath">(n,q)</span> is the group of all <span class="SimpleMath">n × n</span> unitary matrices with entries in <span class="SimpleMath">GF(q^2)</span> and determinant 1.</p>
<p>The purpose of the function is to provide a simple way to formulate a loop over all library groups which contain certain composition factors.</p>
<p><a id="X845419F07BB92867" name="X845419F07BB92867"></a></p>
<h5>50.8-7 <span class="Heading">DisplayInformationPerfectGroups</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayInformationPerfectGroups</code>( <var class="Arg">size</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayInformationPerfectGroups</code>( <var class="Arg">sizenumberpair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">DisplayInformationPerfectGroups</code> displays some invariants of the <var class="Arg">n</var>-th group of order <var class="Arg">size</var> from the perfect groups library.</p>
<p>If no value of <var class="Arg">n</var> has been specified, the invariants will be displayed for all groups of size <var class="Arg">size</var> available in the library.</p>
<p>Alternatively, also a list of length two may be entered as the only argument, with entries <var class="Arg">size</var> and <var class="Arg">n</var>.</p>
<p>The information provided for <span class="SimpleMath">G</span> includes the following items:</p>
<ul>
<li><p>a headline containing the size number <code class="code">[ <var class="Arg">size</var>, <var class="Arg">n</var> ]</code> of <span class="SimpleMath">G</span> in the form <code class="code"><var class="Arg">size</var>.<var class="Arg">n</var></code> (the suffix <code class="code">.<var class="Arg">n</var></code> will be suppressed if, up to isomorphism, <span class="SimpleMath">G</span> is the only perfect group of order <var class="Arg">size</var>),</p>
</li>
<li><p>a message if <span class="SimpleMath">G</span> is simple or quasisimple, i.e., if the factor group of <span class="SimpleMath">G</span> by its centre is simple,</p>
</li>
<li><p>the "description" of the structure of <span class="SimpleMath">G</span> as it is given by Holt and Plesken in <a href="chapBib.html#biBHP89">[HP89]</a> (see below),</p>
</li>
<li><p>the size of the centre of <span class="SimpleMath">G</span> (suppressed, if <span class="SimpleMath">G</span> is simple),</p>
</li>
<li><p>the prime decomposition of the size of <span class="SimpleMath">G</span>,</p>
</li>
<li><p>orbit sizes for a faithful permutation representation of <span class="SimpleMath">G</span> which is provided by the library (see below),</p>
</li>
<li><p>a reference to each occurrence of <span class="SimpleMath">G</span> in the tables of section 5.3 of <a href="chapBib.html#biBHP89">[HP89]</a>. Each of these references consists of a class number and an internal number <span class="SimpleMath">(i,j)</span> under which <span class="SimpleMath">G</span> is listed in that class. For some groups, there is more than one reference because these groups belong to more than one of the classes in the book.</p>
</li>
</ul>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayInformationPerfectGroups( 30720, 3 );</span>
#I Perfect group 30720: A5 ( 2^4 E N 2^1 E 2^4 ) A
#I size = 2^11*3*5 orbit size = 240
#I Holt-Plesken class 1 (9,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayInformationPerfectGroups( 30720, 6 );</span>
#I Perfect group 30720: A5 ( 2^4 x 2^4 ) C N 2^1
#I centre = 2 size = 2^11*3*5 orbit size = 384
#I Holt-Plesken class 1 (9,6)
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayInformationPerfectGroups( Factorial( 8 ) / 2 );</span>
#I Perfect group 20160.1: A5 x L3(2) 2^1
#I centre = 2 size = 2^6*3^2*5*7 orbit sizes = 5 + 16
#I Holt-Plesken class 31 (1,1) (occurs also in class 32)
#I Perfect group 20160.2: A5 2^1 x L3(2)
#I centre = 2 size = 2^6*3^2*5*7 orbit sizes = 7 + 24
#I Holt-Plesken class 31 (1,2) (occurs also in class 32)
#I Perfect group 20160.3: ( A5 x L3(2) ) 2^1
#I centre = 2 size = 2^6*3^2*5*7 orbit size = 192
#I Holt-Plesken class 31 (1,3)
#I Perfect group 20160.4: simple group A8
#I size = 2^6*3^2*5*7 orbit size = 8
#I Holt-Plesken class 26 (0,1)
#I Perfect group 20160.5: simple group L3(4)
#I size = 2^6*3^2*5*7 orbit size = 21
#I Holt-Plesken class 27 (0,1)
</pre></div>
<p><a id="X875C5BE67BAB7F71" name="X875C5BE67BAB7F71"></a></p>
<h5>50.8-8 <span class="Heading">More about the Perfect Groups Library</span></h5>
<p>For any library group <span class="SimpleMath">G</span>, the library files do not only provide a presentation, but, in addition, a list of one or more subgroups <span class="SimpleMath">S_1, ..., S_r</span> of <span class="SimpleMath">G</span> such that there is a faithful permutation representation of <span class="SimpleMath">G</span> of degree <span class="SimpleMath">∑_{i = 1}^r [G:S_i]</span> on the set <span class="SimpleMath">{ S_i g ∣ 1 ≤ i ≤ r, g ∈ G }</span> of the cosets of the <span class="SimpleMath">S_i</span>. This allows one to construct the groups as permutation groups. The function <code class="func">DisplayInformationPerfectGroups</code> (<a href="chap50.html#X845419F07BB92867"><span class="RefLink">50.8-7</span></a>) displays only the available degree. The message</p>
<div class="example"><pre>
orbit size = 8
</pre></div>
<p>in the above example means that the available permutation representation is transitive and of degree 8, whereas the message</p>
<div class="example"><pre>
orbit sizes = 5 + 16
</pre></div>
<p>means that a nontransitive permutation representation is available which acts on two orbits of size 5 and 16 respectively.</p>
<p>The notation used in the "description" of a group is explained in section 5.1.2 of <a href="chapBib.html#biBHP89">[HP89]</a>. We quote the respective page from there:</p>
<p>Within a class <span class="SimpleMath">Q#p</span>, an isomorphism type of groups will be denoted by an ordered pair of integers <span class="SimpleMath">(r,n)</span>, where <span class="SimpleMath">r ≥ 0</span> and <span class="SimpleMath">n > 0</span>. More precisely, the isomorphism types in <span class="SimpleMath">Q # p</span> of order <span class="SimpleMath">p^r |Q|</span> will be denoted by <span class="SimpleMath">(r,1), (r,2), (r,3), ...</span>. Thus <span class="SimpleMath">Q</span> will always get the size number <span class="SimpleMath">(0,1)</span>.</p>
<p>In addition to the symbol <span class="SimpleMath">(r,n)</span>, the groups in <span class="SimpleMath">Q#p</span> will also be given a more descriptive name. The purpose of this is to provide a very rough idea of the structure of the group. The names are derived in the following manner. First of all, the isomorphism classes of irreducible <span class="SimpleMath">F_pQ</span>-modules <span class="SimpleMath">M</span> with <span class="SimpleMath">|Q|.|M| ≤ 10^6</span>, where <span class="SimpleMath">F_p</span> is the field of order <span class="SimpleMath">p</span>, are assigned symbols. These will either be simply <span class="SimpleMath">p^x</span>, where <span class="SimpleMath">x</span> is the dimension of the module, or, if there is more than one isomorphism class of irreducible modules having the same dimension, they will be denoted by<span class="SimpleMath">p^x</span>, <span class="SimpleMath">p^{x'}</span>, etc. The one-dimensional module with trivial <span class="SimpleMath">Q</span>-action will therefore be denoted by <span class="SimpleMath">p^1</span>. These symbols will be listed under the description of <span class="SimpleMath">Q</span>. The group name consists essentially of a list of the composition factors working from the top of the group downwards; hence it always starts with the name of <span class="SimpleMath">Q</span> itself. (This convention is the most convenient in our context, but it is different from that adopted in the ATLAS <a href="chapBib.html#biBCCN85">[CCNPW85]</a>, for example, where composition factors are listed in the reverse order. For example, we denote a group isomorphic to <span class="SimpleMath">SL(2,5)</span> by <span class="SimpleMath">A_5 2^1</span> rather than <span class="SimpleMath">2.A_5</span>.)</p>
<p>Some other symbols are used in the name, in order to give some idea of the relationship between these composition factors, and splitting properties. We shall now list these additional symbols.</p>
<dl>
<dt><strong class="Mark"><span class="SimpleMath">×</span></strong></dt>
<dd><p>between two factors denotes a direct product of <span class="SimpleMath">F_pQ</span>-modules or groups.</p>
</dd>
<dt><strong class="Mark">C</strong></dt>
<dd><p>(for "commutator") between two factors means that the second lies in the commutator subgroup of the first. Similarly, a segment of the form <span class="SimpleMath">(f_1 × f_2) C f_3</span> would mean that the factors <span class="SimpleMath">f_1</span> and <span class="SimpleMath">f_2</span> commute modulo <span class="SimpleMath">f_3</span> and <span class="SimpleMath">f_3</span> lies in <span class="SimpleMath">[f_1,f_2]</span>.</p>
</dd>
<dt><strong class="Mark">A</strong></dt>
<dd><p>(for "abelian") between two factors indicates that the second is in the <span class="SimpleMath">p</span>th power (but not the commutator subgroup) of the first. "A" may also follow the factors, if bracketed.</p>
</dd>
<dt><strong class="Mark">E</strong></dt>
<dd><p>(for "elementary abelian") between two factors indicates that together they generate an elementary abelian group (modulo subsequent factors), but that the resulting <span class="SimpleMath">F_p Q</span>-module extension does not split.</p>
</dd>
<dt><strong class="Mark">N</strong></dt>
<dd><p>(for "nonsplit") before a factor indicates that <span class="SimpleMath">Q</span> (or possibly its covering group) splits down as far at this factor but not over the factor itself. So "<span class="SimpleMath">Q f_1 N f_2</span>" means that the normal subgroup <span class="SimpleMath">f_1 f_2</span> of the group has no complement but, modulo <span class="SimpleMath">f_2</span>, <span class="SimpleMath">f_1</span>, does have a complement.</p>
</dd>
</dl>
<p>Brackets have their obvious meaning. Summarizing, we have:</p>
<dl>
<dt><strong class="Mark"><span class="SimpleMath">×</span></strong></dt>
<dd><p>= direct product;</p>
</dd>
<dt><strong class="Mark">C</strong></dt>
<dd><p>= commutator subgroup;</p>
</dd>
<dt><strong class="Mark">A</strong></dt>
<dd><p>= abelian;</p>
</dd>
<dt><strong class="Mark">E</strong></dt>
<dd><p>= elementary abelian; and</p>
</dd>
<dt><strong class="Mark">N</strong></dt>
<dd><p>= nonsplit.</p>
</dd>
</dl>
<p>Here are some examples.</p>
<dl>
<dt><strong class="Mark">(i)</strong></dt>
<dd><p><span class="SimpleMath">A_5 (2^4 E 2^1 E 2^4) A</span> means that the pairs <span class="SimpleMath">2^4 E 2^1</span> and <span class="SimpleMath">2^1 E 2^4</span> are both elementary abelian of exponent 4.</p>
</dd>
<dt><strong class="Mark">(ii)</strong></dt>
<dd><p><span class="SimpleMath">A_5 (2^4 E 2^1 A) C 2^1</span> means that <span class="SimpleMath">O_2(G)</span> is of symplectic type <span class="SimpleMath">2^{1+5}</span>, with Frattini factor group of type <span class="SimpleMath">2^4 E 2^1</span>. The "A" after the <span class="SimpleMath">2^1</span> indicates that <span class="SimpleMath">G</span> has a central cyclic subgroup <span class="SimpleMath">2^1 A 2^1</span> of order 4.</p>
</dd>
<dt><strong class="Mark">(iii)</strong></dt>
<dd><p><span class="SimpleMath">L_3(2) ((2^1 E) × ( N 2^3 E 2^{3'} A) C) 2^{3'}</span> means that the <span class="SimpleMath">2^{3'}</span> factor at the bottom lies in the commutator subgroup of the pair <span class="SimpleMath">2^3 E 2^{3'}</span> in the middle, but the lower pair <span class="SimpleMath">2^{3'} A 2^{3'}</span> is abelian of exponent 4. There is also a submodule <span class="SimpleMath">2^1 E 2^{3'}</span>, and the covering group <span class="SimpleMath">L_3(2) 2^1</span> of <span class="SimpleMath">L_3(2)</span> does not split over the <span class="SimpleMath">2^3</span> factor. (Since <span class="SimpleMath">G</span> is perfect, it goes without saying that the extension <span class="SimpleMath">L_3(2) 2^1</span> cannot split itself.)</p>
</dd>
</dl>
<p>We must stress that this notation does not always succeed in being precise or even unambiguous, and the reader is free to ignore it if it does not seem helpful.</p>
<p>If such a group description has been given in the book for <span class="SimpleMath">G</span> (and, in fact, this is the case for most of the library groups), it is displayed by <code class="func">DisplayInformationPerfectGroups</code> (<a href="chap50.html#X845419F07BB92867"><span class="RefLink">50.8-7</span></a>). Otherwise the function provides a less explicit description of the (in these cases unique) Holt-Plesken class to which <span class="SimpleMath">G</span> belongs, together with a serial number if this is necessary to make it unique.</p>
<p><a id="X7AE00EA7791F2574" name="X7AE00EA7791F2574"></a></p>
<h4>50.9 <span class="Heading">Primitive Permutation Groups</span></h4>
<p><strong class="pkg">GAP</strong> contains a library of primitive permutation groups which includes, up to permutation isomorphism (i.e., up to conjugacy in the corresponding symmetric group), all primitive permutation groups of degree <span class="SimpleMath">< 2500</span>, calculated in <a href="chapBib.html#biBRoneyDougal05">[Ron05]</a>, in particular,</p>
<ul>
<li><p>the primitive permutation groups up to degree 50, calculated by C. Sims,</p>
</li>
<li><p>the primitive groups with insoluble socles of degree <span class="SimpleMath">< 1000</span> as calculated in <a href="chapBib.html#biBDixonMortimer88">[DM88]</a>,</p>
</li>
<li><p>the solvable (hence affine) primitive permutation groups of degree <span class="SimpleMath">< 256</span> as calculated by M. Short <a href="chapBib.html#biBSho92">[Sho92]</a>,</p>
</li>
<li><p>some insolvable affine primitive permutation groups of degree <span class="SimpleMath">< 256</span> as calculated in <a href="chapBib.html#biBTheissen97">[The97]</a>.</p>
</li>
<li><p>The solvable primitive groups of degree up to <span class="SimpleMath">999</span> as calculated in <a href="chapBib.html#biBEickHoefling02">[EH03]</a>.</p>
</li>
<li><p>The primitive groups of affine type of degree up to <span class="SimpleMath">999</span> as calculated in <a href="chapBib.html#biBRoneyDougal02">[RU03]</a>.</p>
</li>
</ul>
<p>Not all groups are named, those which do have names use ATLAS notation. Not all names are necessary unique!</p>
<p>The list given in <a href="chapBib.html#biBRoneyDougal05">[Ron05]</a> is believed to be complete, correcting various omissions in <a href="chapBib.html#biBDixonMortimer88">[DM88]</a>, <a href="chapBib.html#biBSho92">[Sho92]</a> and <a href="chapBib.html#biBTheissen97">[The97]</a>.</p>
<p>In detail, we guarantee the following properties for this and further versions (but <em>not</em> versions which came before <strong class="pkg">GAP</strong> 4.2) of the library:</p>
<ul>
<li><p>All groups in the library are primitive permutation groups of the indicated degree.</p>
</li>
<li><p>The positions of the groups in the library are stable. That is <code class="code">PrimitiveGroup(<var class="Arg">n</var>,<var class="Arg">nr</var>)</code> will always give you a permutation isomorphic group. Note however that we do not guarantee to keep the chosen <span class="SimpleMath">S_n</span>-representative, the generating set or the name for eternity.</p>
</li>
<li><p>Different groups in the library are not conjugate in <span class="SimpleMath">S_n</span>.</p>
</li>
<li><p>If a group in the library has a primitive subgroup with the same socle, this group is in the library as well.</p>
</li>
</ul>
<p>(Note that the arrangement of groups is not guaranteed to be in increasing size, though it holds for many degrees.)</p>
<p>The selection functions (see <a href="chap50.html#X82676ED5826E9E2E"><span class="RefLink">50.5</span></a>) for the primitive groups library are <code class="code">AllPrimitiveGroups</code> and <code class="code">OnePrimitiveGroup</code>. They obtain the following properties from the database without having to compute them anew:</p>
<p><code class="func">NrMovedPoints</code> (<a href="chap42.html#X85E7B1E28430F49E"><span class="RefLink">42.3-4</span></a>), <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>), <code class="func">Transitivity</code> (<a href="chap41.html#X8295D733796B7A37"><span class="RefLink">41.10-2</span></a>), <code class="func">ONanScottType</code> (<a href="chap43.html#X7E50211A7B92455F"><span class="RefLink">43.5-1</span></a>), <code class="func">IsSimpleGroup</code> (<a href="chap39.html#X7A6685D7819AEC32"><span class="RefLink">39.15-10</span></a>), <code class="func">IsSolvableGroup</code> (<a href="chap39.html#X809C78D5877D31DF"><span class="RefLink">39.15-6</span></a>), and <code class="func">SocleTypePrimitiveGroup</code> (<a href="chap43.html#X7E89A46A86A3F4A2"><span class="RefLink">43.5-2</span></a>).</p>
<p>(Note, that for groups of degree up to 2499, O'Nan-Scott types 4a, 4b and 5 cannot occur.)</p>
<p><a id="X7BCEA0C57B6D9F42" name="X7BCEA0C57B6D9F42"></a></p>
<h5>50.9-1 PrimitiveGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimitiveGroup</code>( <var class="Arg">deg</var>, <var class="Arg">nr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the primitive permutation group of degree <var class="Arg">deg</var> with number <var class="Arg">nr</var> from the list.</p>
<p>The arrangement of the groups differs from the arrangement of primitive groups in the list of C. Sims, which was used in <strong class="pkg">GAP</strong> 3. See <code class="func">SimsNo</code> (<a href="chap50.html#X790D50447ABDF7EE"><span class="RefLink">50.10-2</span></a>).</p>
<p><a id="X8564FECC8477F199" name="X8564FECC8477F199"></a></p>
<h5>50.9-2 NrPrimitiveGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrPrimitiveGroups</code>( <var class="Arg">deg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the number of primitive permutation groups of degree <var class="Arg">deg</var> in the library.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrPrimitiveGroups(25);</span>
28
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveGroup(25,19);</span>
5^2:((Q(8):3)'4)
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveGroup(25,20);</span>
ASL(2, 5)
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveGroup(25,22);</span>
AGL(2, 5)
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveGroup(25,23);</span>
(A(5) x A(5)):2
</pre></div>
<p><a id="X7B1D4C0483A7F444" name="X7B1D4C0483A7F444"></a></p>
<h5>50.9-3 PrimitiveGroupsIterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimitiveGroupsIterator</code>( <var class="Arg">attr1</var>, <var class="Arg">val1</var>, <var class="Arg">attr2</var>, <var class="Arg">val2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an iterator through <code class="code">AllPrimitiveGroups(<var class="Arg">attr1</var>,<var class="Arg">val1</var>,<var class="Arg">attr2</var>,<var class="Arg">val2</var>,...)</code> without creating all these groups at the same time.</p>
<p><a id="X81329B9B7F5FF8DE" name="X81329B9B7F5FF8DE"></a></p>
<h5>50.9-4 COHORTS_PRIMITIVE_GROUPS</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ COHORTS_PRIMITIVE_GROUPS</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>In <a href="chapBib.html#biBDixonMortimer88">[DM88]</a> the primitive groups are sorted in "cohorts" according to their socle. For each degree, the variable <code class="func">COHORTS_PRIMITIVE_GROUPS</code> contains a list of the cohorts for the primitive groups of this degree. Each cohort is represented by a list of length 2, the first entry specifies the socle type (see <code class="func">SocleTypePrimitiveGroup</code> (<a href="chap43.html#X7E89A46A86A3F4A2"><span class="RefLink">43.5-2</span></a>)), the second entry listing the index numbers of the groups in this degree.</p>
<p>For example in degree 49, we have four cohorts with socles <span class="SimpleMath">(ℤ / 7 ℤ)^2</span>, <span class="SimpleMath">L_2(7)^2</span>, <span class="SimpleMath">A_7^2</span> and <span class="SimpleMath">A_49</span> respectively. the group <code class="code">PrimitiveGroup(49,36)</code>, which is isomorphic to <span class="SimpleMath">(A_7 × A_7):2^2</span>, lies in the third cohort with socle <span class="SimpleMath">(A_7 × A_7)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">COHORTS_PRIMITIVE_GROUPS[49];</span>
[ [ rec( parameter := 7, series := "Z", width := 2 ),
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33 ] ],
[ rec( parameter := [ 2, 7 ], series := "L", width := 2 ), [ 34 ] ],
[ rec( parameter := 7, series := "A", width := 2 ),
[ 35, 36, 37, 38 ] ],
[ rec( parameter := 49, series := "A", width := 1 ), [ 39, 40 ] ] ]
</pre></div>
<p><a id="X7DA239CC848F6CAE" name="X7DA239CC848F6CAE"></a></p>
<h4>50.10 <span class="Heading">Index numbers of primitive groups</span></h4>
<p><a id="X870400597FD4E392" name="X870400597FD4E392"></a></p>
<h5>50.10-1 PrimitiveIdentification</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimitiveIdentification</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a primitive permutation group for which an <span class="SimpleMath">S_n</span>-conjugate exists in the library of primitive permutation groups (see <a href="chap50.html#X7AE00EA7791F2574"><span class="RefLink">50.9</span></a>), this attribute returns the index position. That is <var class="Arg">G</var> is conjugate to <code class="code">PrimitiveGroup(NrMovedPoints(<var class="Arg">G</var>),PrimitiveIdentification(<var class="Arg">G</var>))</code>.</p>
<p>Methods only exist if the primitive groups library is installed.</p>
<p>Note: As this function uses the primitive groups library, the result is only guaranteed to the same extent as this library. If it is incomplete, <code class="code">PrimitiveIdentification</code> might return an existing index number for a group not in the library.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveIdentification(Group((1,2),(1,2,3)));</span>
2
</pre></div>
<p><a id="X790D50447ABDF7EE" name="X790D50447ABDF7EE"></a></p>
<h5>50.10-2 SimsNo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimsNo</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">G</var> is a primitive group obtained by <code class="func">PrimitiveGroup</code> (<a href="chap50.html#X7BCEA0C57B6D9F42"><span class="RefLink">50.9-1</span></a>) (respectively one of the selection functions) this attribute contains the number of the isomorphic group in the original list of C. Sims. (This is the arrangement as it was used in <strong class="pkg">GAP</strong> 3.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=PrimitiveGroup(25,2);</span>
5^2:S(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">SimsNo(g);</span>
3
</pre></div>
<p>As mentioned in the previous section, the index numbers of primitive groups in <strong class="pkg">GAP</strong> are guaranteed to remain stable. (Thus, missing groups will be added to the library at the end of each degree.) In particular, it is safe to refer to a primitive group of type <var class="Arg">deg</var>, <var class="Arg">nr</var> in the <strong class="pkg">GAP</strong> library.</p>
<p><a id="X784820DA86D0E6F4" name="X784820DA86D0E6F4"></a></p>
<h5>50.10-3 PRIMITIVE_INDICES_MAGMA</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PRIMITIVE_INDICES_MAGMA</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>The system <strong class="pkg">Magma</strong> also provides a list of primitive groups (see <a href="chapBib.html#biBRoneyDougal02">[RU03]</a>). For historical reasons, its indexing up to degree 999 differs from the one used by <strong class="pkg">GAP</strong>. The variable <code class="func">PRIMITIVE_INDICES_MAGMA</code> can be used to obtain this correspondence. The magma index number of the <strong class="pkg">GAP</strong> group <code class="code">PrimitiveGroup(<var class="Arg">deg</var>,<var class="Arg">nr</var>)</code> is stored in the entry <code class="code">PRIMITIVE_INDICES_MAGMA[<var class="Arg">deg</var>][<var class="Arg">nr</var>]</code>, for degree at most 999.</p>
<p>Vice versa, the group of degree <var class="Arg">deg</var> with <strong class="pkg">Magma</strong> index number <var class="Arg">nr</var> has the <strong class="pkg">GAP</strong> index</p>
<p><code class="code">Position(PRIMITIVE_INDICES_MAGMA[<var class="Arg">deg</var>],<var class="Arg">nr</var>)</code>, in particular it can be obtained by the <strong class="pkg">GAP</strong> command</p>
<p><code class="code">PrimitiveGroup(<var class="Arg">deg</var>,Position(PRIMITIVE_INDICES_MAGMA[<var class="Arg">deg</var>],<var class="Arg">nr</var>));</code></p>
<p><a id="X82FD673384BF353B" name="X82FD673384BF353B"></a></p>
<h4>50.11 <span class="Heading">Irreducible Solvable Matrix Groups</span></h4>
<p><a id="X7DF4B4D683A727E8" name="X7DF4B4D683A727E8"></a></p>
<h5>50.11-1 IrreducibleSolvableGroupMS</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IrreducibleSolvableGroupMS</code>( <var class="Arg">n</var>, <var class="Arg">p</var>, <var class="Arg">i</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a representative of the <var class="Arg">i</var>-th conjugacy class of irreducible solvable subgroup of GL(<var class="Arg">n</var>, <var class="Arg">p</var>), where <var class="Arg">n</var> is an integer <span class="SimpleMath">> 1</span>, <var class="Arg">p</var> is a prime, and <span class="SimpleMath"><var class="Arg">p</var>^<var class="Arg">n</var> < 256</span>.</p>
<p>The numbering of the representatives should be considered arbitrary. However, it is guaranteed that the <var class="Arg">i</var>-th group on this list will lie in the same conjugacy class in all future versions of <strong class="pkg">GAP</strong>, unless two (or more) groups on the list are discovered to be duplicates, in which case <code class="func">IrreducibleSolvableGroupMS</code> will return <code class="keyw">fail</code> for all but one of the duplicates.</p>
<p>For values of <var class="Arg">n</var>, <var class="Arg">p</var>, and <var class="Arg">i</var> admissible to <code class="func">IrreducibleSolvableGroup</code> (<a href="chap50.html#X816FF4DD8267B4A7"><span class="RefLink">50.11-6</span></a>), <code class="func">IrreducibleSolvableGroupMS</code> returns a representative of the same conjugacy class of subgroups of GL(<var class="Arg">n</var>, <var class="Arg">p</var>) as <code class="func">IrreducibleSolvableGroup</code> (<a href="chap50.html#X816FF4DD8267B4A7"><span class="RefLink">50.11-6</span></a>). Note that it currently adds two more groups (missing from the original list by Mark Short) for <var class="Arg">n</var> <span class="SimpleMath">= 2</span>, <var class="Arg">p</var> <span class="SimpleMath">= 13</span>.</p>
<p><a id="X836AEF4A7E494724" name="X836AEF4A7E494724"></a></p>
<h5>50.11-2 NumberIrreducibleSolvableGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumberIrreducibleSolvableGroups</code>( <var class="Arg">n</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns the number of conjugacy classes of irreducible solvable subgroup of GL(<var class="Arg">n</var>, <var class="Arg">p</var>).</p>
<p><a id="X7DAC64F17C8B49A2" name="X7DAC64F17C8B49A2"></a></p>
<h5>50.11-3 AllIrreducibleSolvableGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllIrreducibleSolvableGroups</code>( <var class="Arg">func1</var>, <var class="Arg">val1</var>, <var class="Arg">func2</var>, <var class="Arg">val2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a list of conjugacy class representatives <span class="SimpleMath">G</span> of matrix groups over a prime field such that <span class="SimpleMath">f(G) = v</span> or <span class="SimpleMath">f(G) ∈ v</span>, for all pairs <span class="SimpleMath">(f,v)</span> in (<var class="Arg">func1</var>, <var class="Arg">val1</var>), (<var class="Arg">func2</var>, <var class="Arg">val2</var>), <span class="SimpleMath">...</span>. The following possibilities for the functions <span class="SimpleMath">f</span> are particularly efficient, because the values can be read off the information in the data base: <code class="code">DegreeOfMatrixGroup</code> (or <code class="func">Dimension</code> (<a href="chap57.html#X7E6926C6850E7C4E"><span class="RefLink">57.3-3</span></a>) or <code class="func">DimensionOfMatrixGroup</code> (<a href="chap44.html#X7E55258C783C50CA"><span class="RefLink">44.2-1</span></a>)) for the linear degree, <code class="func">Characteristic</code> (<a href="chap31.html#X81278E53800BF64D"><span class="RefLink">31.10-1</span></a>) for the field characteristic, <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>), <code class="code">IsPrimitiveMatrixGroup</code> (or <code class="code">IsLinearlyPrimitive</code>), and <code class="code">MinimalBlockDimension</code>>.</p>
<p><a id="X844E60B87FC48D1B" name="X844E60B87FC48D1B"></a></p>
<h5>50.11-4 OneIrreducibleSolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneIrreducibleSolvableGroup</code>( <var class="Arg">func1</var>, <var class="Arg">val1</var>, <var class="Arg">func2</var>, <var class="Arg">val2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns one solvable subgroup <span class="SimpleMath">G</span> of a matrix group over a prime field such that <span class="SimpleMath">f(G) = v</span> or <span class="SimpleMath">f(G) ∈ v</span>, for all pairs <span class="SimpleMath">(f,v)</span> in (<var class="Arg">func1</var>, <var class="Arg">val1</var>), (<var class="Arg">func2</var>, <var class="Arg">val2</var>), <span class="SimpleMath">...</span>. The following possibilities for the functions <span class="SimpleMath">f</span> are particularly efficient, because the values can be read off the information in the data base: <code class="code">DegreeOfMatrixGroup</code> (or <code class="func">Dimension</code> (<a href="chap57.html#X7E6926C6850E7C4E"><span class="RefLink">57.3-3</span></a>) or <code class="func">DimensionOfMatrixGroup</code> (<a href="chap44.html#X7E55258C783C50CA"><span class="RefLink">44.2-1</span></a>)) for the linear degree, <code class="func">Characteristic</code> (<a href="chap31.html#X81278E53800BF64D"><span class="RefLink">31.10-1</span></a>) for the field characteristic, <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>), <code class="code">IsPrimitiveMatrixGroup</code> (or <code class="code">IsLinearlyPrimitive</code>), and <code class="code">MinimalBlockDimension</code>>.</p>
<p><a id="X81B11EE77EFA745E" name="X81B11EE77EFA745E"></a></p>
<h5>50.11-5 PrimitiveIndexIrreducibleSolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimitiveIndexIrreducibleSolvableGroup</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>This variable provides a way to get from irreducible solvable groups to primitive groups and vice versa. For the group <span class="SimpleMath">G</span> = <code class="code">IrreducibleSolvableGroup( <var class="Arg">n</var>, <var class="Arg">p</var>, <var class="Arg">k</var> )</code> and <span class="SimpleMath">d = p^n</span>, the entry <code class="code">PrimitiveIndexIrreducibleSolvableGroup[d][i]</code> gives the index number of the semidirect product <span class="SimpleMath">p^n:G</span> in the library of primitive groups.</p>
<p>Searching for an index in this list with <code class="func">Position</code> (<a href="chap21.html#X79975EC6783B4293"><span class="RefLink">21.16-1</span></a>) gives the translation in the other direction.</p>
<p><a id="X816FF4DD8267B4A7" name="X816FF4DD8267B4A7"></a></p>
<h5>50.11-6 IrreducibleSolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IrreducibleSolvableGroup</code>( <var class="Arg">n</var>, <var class="Arg">p</var>, <var class="Arg">i</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function is obsolete, because for <var class="Arg">n</var> <span class="SimpleMath">= 2</span>, <var class="Arg">p</var> <span class="SimpleMath">= 13</span>, two groups were missing from the underlying database. It has been replaced by the function <code class="func">IrreducibleSolvableGroupMS</code> (<a href="chap50.html#X7DF4B4D683A727E8"><span class="RefLink">50.11-1</span></a>). Please note that the latter function does not guarantee any ordering of the groups in the database. However, for values of <var class="Arg">n</var>, <var class="Arg">p</var>, and <var class="Arg">i</var> admissible to <code class="func">IrreducibleSolvableGroup</code>, <code class="func">IrreducibleSolvableGroupMS</code> (<a href="chap50.html#X7DF4B4D683A727E8"><span class="RefLink">50.11-1</span></a>) returns a representative of the same conjugacy class of subgroups of GL(<var class="Arg">n</var>, <var class="Arg">p</var>) as <code class="func">IrreducibleSolvableGroup</code> did before.</p>
<p><a id="X7873506D873EDB95" name="X7873506D873EDB95"></a></p>
<h4>50.12 <span class="Heading">Irreducible Maximal Finite Integral Matrix Groups</span></h4>
<p>A library of irreducible maximal finite integral matrix groups is provided with <strong class="pkg">GAP</strong>. It contains <span class="SimpleMath">ℚ</span>-class representatives for all of these groups of dimension at most 31, and <span class="SimpleMath">ℤ</span>-class representatives for those of dimension at most 11 or of dimension 13, 17, 19, or 23.</p>
<p>The groups provided in this library have been determined by Wilhelm Plesken, partially as joint work with Michael Pohst, or by members of his institute (Lehrstuhl B für Mathematik, RWTH Aachen). In particular, the data for the groups of dimensions 2 to 9 have been taken from the output of computer calculations which they performed in 1979 (see <a href="chapBib.html#biBPP77">[PP77]</a>, <a href="chapBib.html#biBPP80">[PP80]</a>). The <span class="SimpleMath">ℤ</span>-class representatives of the groups of dimension 10 have been determined and computed by Bernd Souvignier (<a href="chapBib.html#biBSou94">[Sou94]</a>), and those of dimensions 11, 13, and 17 have been recomputed for this library from the circulant Gram matrices given in <a href="chapBib.html#biBPle85">[Ple85]</a>, using the stand-alone programs for the computation of short vectors and Bravais groups which have been developed in Plesken's institute. The <span class="SimpleMath">ℤ</span>-class representatives of the groups of dimensions 19 and 23 had already been determined in <a href="chapBib.html#biBPle85">[Ple85]</a>. Gabriele Nebe has recomputed them for us. Her main contribution to this library, however, is that she has determined and computed the <span class="SimpleMath">ℚ</span>-class representatives of the groups of non-prime dimensions between 12 and 24 and the groups of dimensions 25 to 31 (see <a href="chapBib.html#biBPN95">[PN95]</a>, <a href="chapBib.html#biBNP95">[NP95]</a>, <a href="chapBib.html#biBNeb95">[Neb95]</a>, <a href="chapBib.html#biBNeb96">[Neb96]</a>).</p>
<p>The library has been brought into <strong class="pkg">GAP</strong> format by Volkmar Felsch. He has applied several <strong class="pkg">GAP</strong> routines to check certain consistency of the data. However, the credit and responsibility for the lists remain with the authors. We are grateful to Wilhelm Plesken, Gabriele Nebe, and Bernd Souvignier for supplying their results to <strong class="pkg">GAP</strong>.</p>
<p>In the preceding acknowledgement, we used some notations that will also be needed in the sequel. We first define these.</p>
<p>Any integral matrix group <span class="SimpleMath">G</span> of dimension <span class="SimpleMath">n</span> is a subgroup of <span class="SimpleMath">GL_n(ℤ)</span> as well as of <span class="SimpleMath">GL_n(ℚ)</span> and hence lies in some conjugacy class of integral matrix groups under <span class="SimpleMath">GL_n(ℤ)</span> and also in some conjugacy class of rational matrix groups under <span class="SimpleMath">GL_n(ℚ)</span>. As usual, we call these classes the <span class="SimpleMath">ℤ</span>-class and the <span class="SimpleMath">ℚ</span>-class of <span class="SimpleMath">G</span>, respectively. Note that any conjugacy class of subgroups of <span class="SimpleMath">GL_n(ℚ)</span> contains at least one <span class="SimpleMath">ℤ</span>-class of subgroups of <span class="SimpleMath">GL_n(ℤ)</span> and hence can be considered as the <span class="SimpleMath">ℚ</span>-class of some integral matrix group.</p>
<p>In the context of this library we are only concerned with <span class="SimpleMath">ℤ</span>-classes and <span class="SimpleMath">ℚ</span>-classes of subgroups of <span class="SimpleMath">GL_n(ℤ)</span> which are irreducible and maximal finite in <span class="SimpleMath">GL_n(ℤ)</span> (we will call them <em>i.m.f.</em> subgroups of <span class="SimpleMath">GL_n(ℤ)</span>). We can distinguish two types of these groups:</p>
<p>First, there are those i.m.f. subgroups of <span class="SimpleMath">GL_n(ℤ)</span> which are also maximal finite subgroups of <span class="SimpleMath">GL_n(ℚ)</span>. Let us denote the set of their <span class="SimpleMath">ℚ</span>-classes by <span class="SimpleMath">Q_1(n)</span>. It is clear from the above remark that <span class="SimpleMath">Q_1(n)</span> just consists of the <span class="SimpleMath">ℚ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_n(ℚ)</span>.</p>
<p>Secondly, there is the set <span class="SimpleMath">Q_2(n)</span> of the <span class="SimpleMath">ℚ</span>-classes of the remaining i.m.f. subgroups of <span class="SimpleMath">GL_n(ℤ)</span>, i.e., of those which are not maximal finite subgroups of <span class="SimpleMath">GL_n(ℚ)</span>. For any such group <span class="SimpleMath">G</span>, say, there is at least one class <span class="SimpleMath">C ∈ Q_1(n)</span> such that <span class="SimpleMath">G</span> is conjugate under <span class="SimpleMath">ℚ</span> to a proper subgroup of some group <span class="SimpleMath">H ∈ C</span>. In fact, the class <span class="SimpleMath">C</span> is uniquely determined for any group <span class="SimpleMath">G</span> occurring in the library (though there seems to be no reason to assume that this property should hold in general). Hence we may call <span class="SimpleMath">C</span> the <em>rational i.m.f. class</em> of <span class="SimpleMath">G</span>. Finally, we will denote the number of classes in <span class="SimpleMath">Q_1(n)</span> and <span class="SimpleMath">Q_2(n)</span> by <span class="SimpleMath">q_1(n)</span> and <span class="SimpleMath">q_2(n)</span>, respectively.</p>
<p>As an example, let us consider the case <span class="SimpleMath">n = 4</span>. There are 6 <span class="SimpleMath">ℤ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_4(ℤ)</span> with representative subgroups <span class="SimpleMath">G_1, ..., G_6</span> of isomorphism types <span class="SimpleMath">G_1 ≅ W(F_4)</span>, <span class="SimpleMath">G_2 ≅ D_12 ≀ C_2</span>, <span class="SimpleMath">G_3 ≅ G_4 ≅ C_2 × S_5</span>, <span class="SimpleMath">G_5 ≅ W(B_4)</span>, and <span class="SimpleMath">G_6 ≅ (D_12</span><code class="code">Y</code><span class="SimpleMath">D_12) : C_2</span>. The corresponding <span class="SimpleMath">ℚ</span>-classes, <span class="SimpleMath">R_1, ..., R_6</span>, say, are pairwise different except that <span class="SimpleMath">R_3</span> coincides with <span class="SimpleMath">R_4</span>. The groups <span class="SimpleMath">G_1</span>, <span class="SimpleMath">G_2</span>, and <span class="SimpleMath">G_3</span> are i.m.f. subgroups of <span class="SimpleMath">GL_4(ℚ)</span>, but <span class="SimpleMath">G_5</span> and <span class="SimpleMath">G_6</span> are not because they are conjugate under <span class="SimpleMath">GL_4(ℚ)</span> to proper subgroups of <span class="SimpleMath">G_1</span> and <span class="SimpleMath">G_2</span>, respectively. So we have <span class="SimpleMath">Q_1(4) = { R_1, R_2, R_3 }</span>, <span class="SimpleMath">Q_2(4) = { R_5, R_6 }</span>, <span class="SimpleMath">q_1(4) = 3</span>, and <span class="SimpleMath">q_2(4) = 2</span>.</p>
<p>The <span class="SimpleMath">q_1(n)</span> <span class="SimpleMath">ℚ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_n(ℚ)</span> have been determined for each dimension <span class="SimpleMath">n ≤ 31</span>. The current <strong class="pkg">GAP</strong> library provides integral representative groups for all these classes. Moreover, all <span class="SimpleMath">ℤ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_n(ℤ)</span> are known for <span class="SimpleMath">n ≤ 11</span> and for <span class="SimpleMath">n ∈ {13,17,19,23}</span>. For these dimensions, the library offers integral representative groups for all <span class="SimpleMath">ℚ</span>-classes in <span class="SimpleMath">Q_1(n)</span> and <span class="SimpleMath">Q_2(n)</span> as well as for all <span class="SimpleMath">ℤ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_n(ℤ)</span>.</p>
<p>Any group <span class="SimpleMath">G</span> of dimension <span class="SimpleMath">n</span> given in the library is represented as the automorphism group <span class="SimpleMath">G = Aut(F,L) = { g ∈ GL_n(ℤ) ∣ Lg = L, g F g^tr = F }</span> of a positive definite symmetric <span class="SimpleMath">n × n</span> matrix <span class="SimpleMath">F ∈ ℤ^{n × n}</span> on an <span class="SimpleMath">n</span>-dimensional lattice <span class="SimpleMath">L ≅ ℤ^{1 × n}</span> (for details see e.g. <a href="chapBib.html#biBPN95">[PN95]</a>). <strong class="pkg">GAP</strong> provides for <span class="SimpleMath">G</span> a list of matrix generators and the <em>Gram matrix</em> <span class="SimpleMath">F</span>.</p>
<p>The positive definite quadratic form defined by <span class="SimpleMath">F</span> defines a <em>norm</em> <span class="SimpleMath">v F v^tr</span> for each vector <span class="SimpleMath">v ∈ L</span>, and there is only a finite set of vectors of minimal norm. These vectors are often simply called the <em>short vectors</em>. Their set splits into orbits under <span class="SimpleMath">G</span>, and <span class="SimpleMath">G</span> being irreducible acts faithfully on each of these orbits by multiplication from the right. <strong class="pkg">GAP</strong> provides for each of these orbits the orbit size and a representative vector.</p>
<p>Like most of the other <strong class="pkg">GAP</strong> libraries, the library of i.m.f. integral matrix groups supplies an extraction function, <code class="code">ImfMatrixGroup</code>. However, as the library involves only 525 different groups, there is no need for a selection or an example function. Instead, there are two functions, <code class="func">ImfInvariants</code> (<a href="chap50.html#X8604A2167B2E8434"><span class="RefLink">50.12-3</span></a>) and <code class="func">DisplayImfInvariants</code> (<a href="chap50.html#X8705F64B7E19DDC7"><span class="RefLink">50.12-2</span></a>), which provide some <span class="SimpleMath">ℤ</span>-class invariants that can be extracted from the library without actually constructing the representative groups themselves. The difference between these two functions is that the latter one displays the resulting data in some easily readable format, whereas the first one returns them as record components so that you can properly access them.</p>
<p>We shall give an individual description of each of the library functions, but first we would like to insert a short remark concerning their names: Any self-explaining name of a function handling <em>irreducible maximal finite integral matrix groups</em> would have to include this term in full length and hence would grow extremely long. Therefore we have decided to use the abbreviation <code class="code">Imf</code> instead in order to restrict the names to some reasonable length.</p>
<p>The first three functions can be used to formulate loops over the classes.</p>
<p><a id="X8693FD647EF3C53B" name="X8693FD647EF3C53B"></a></p>
<h5>50.12-1 ImfNumberQQClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImfNumberQQClasses</code>( <var class="Arg">dim</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImfNumberQClasses</code>( <var class="Arg">dim</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImfNumberZClasses</code>( <var class="Arg">dim</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">ImfNumberQQClasses</code> returns the number <span class="SimpleMath">q_1(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> of <span class="SimpleMath">ℚ</span>-classes of i.m.f. rational matrix groups of dimension <var class="Arg">dim</var>. Valid values of <var class="Arg">dim</var> are all positive integers up to 31.</p>
<p>Note: In order to enable you to loop just over the classes belonging to <span class="SimpleMath">Q_1(</span><var class="Arg">dim</var><span class="SimpleMath">)</span>, we have arranged the list of <span class="SimpleMath">ℚ</span>-classes of dimension <var class="Arg">dim</var> for any dimension <var class="Arg">dim</var> in the library such that, whenever the classes of <span class="SimpleMath">Q_2(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> are known, too, i.e., in the cases <span class="SimpleMath">dim ≤ 11</span> or <span class="SimpleMath">dim ∈ {13,17,19,23}</span>, the classes of <span class="SimpleMath">Q_1(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> precede those of <span class="SimpleMath">Q_2(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> and hence are numbered from 1 to <span class="SimpleMath">q_1(</span><var class="Arg">dim</var><span class="SimpleMath">)</span>.</p>
<p><code class="code">ImfNumberQClasses</code> returns the number of <span class="SimpleMath">ℚ</span>-classes of groups of dimension <var class="Arg">dim</var> which are available in the library. If <span class="SimpleMath">dim ≤ 11</span> or <span class="SimpleMath">dim ∈ {13,17,19,23}</span>, this is the number <span class="SimpleMath">q_1(</span><var class="Arg">dim</var><span class="SimpleMath">) + q_2(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> of <span class="SimpleMath">ℚ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_dim(ℤ)</span>. Otherwise, it is just the number <span class="SimpleMath">q_1(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> of <span class="SimpleMath">ℚ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_dim(ℚ)</span>. Valid values of <var class="Arg">dim</var> are all positive integers up to 31.</p>
<p><code class="func">ImfNumberZClasses</code> returns the number of <span class="SimpleMath">ℤ</span>-classes in the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class of i.m.f. integral matrix groups of dimension <var class="Arg">dim</var>. Valid values of <var class="Arg">dim</var> are all positive integers up to 11 and all primes up to 23.</p>
<p><a id="X8705F64B7E19DDC7" name="X8705F64B7E19DDC7"></a></p>
<h5>50.12-2 DisplayImfInvariants</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayImfInvariants</code>( <var class="Arg">dim</var>, <var class="Arg">q</var>[, <var class="Arg">z</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">DisplayImfInvariants</code> displays the following <span class="SimpleMath">ℤ</span>-class invariants of the groups in the <var class="Arg">z</var>-th <span class="SimpleMath">ℤ</span>-class in the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class of i.m.f. integral matrix groups of dimension <var class="Arg">dim</var>:</p>
<ul>
<li><p>its <span class="SimpleMath">ℤ</span>-class number in the form <var class="Arg">dim</var>.<var class="Arg">q</var>.<var class="Arg">z</var>, if <var class="Arg">dim</var> is at most 11 or a prime at most 23, or its <span class="SimpleMath">ℚ</span>-class number in the form <var class="Arg">dim</var>.<var class="Arg">q</var>, else,</p>
</li>
<li><p>a message if the group is solvable,</p>
</li>
<li><p>the size of the group,</p>
</li>
<li><p>the isomorphism type of the group,</p>
</li>
<li><p>the elementary divisors of the associated quadratic form,</p>
</li>
<li><p>the sizes of the orbits of short vectors (these sizes are the degrees of the faithful permutation representations which you may construct using the functions <code class="func">IsomorphismPermGroup</code> (<a href="chap50.html#X84BF34B27CD5E85C"><span class="RefLink">50.12-5</span></a>) or <code class="func">IsomorphismPermGroupImfGroup</code> (<a href="chap50.html#X7CEDB6CE7BAC4518"><span class="RefLink">50.12-6</span></a>) below),</p>
</li>
<li><p>the norm of the associated short vectors,</p>
</li>
<li><p>only in case that the group is not an i.m.f. group in <span class="SimpleMath">GL_n(ℚ)</span>: an appropriate message, including the <span class="SimpleMath">ℚ</span>-class number of the corresponding rational i.m.f. class.</p>
</li>
</ul>
<p>If you specify the value 0 for any of the parameters <var class="Arg">dim</var>, <var class="Arg">q</var>, or <var class="Arg">z</var>, the command will loop over all available dimensions, <span class="SimpleMath">ℚ</span>-classes of given dimension, or <span class="SimpleMath">ℤ</span>-classes within the given <span class="SimpleMath">ℚ</span>-class, respectively. Otherwise, the values of the arguments must be in range. A value <var class="Arg">z</var> <span class="SimpleMath">≠ 1</span> must not be specified if the <span class="SimpleMath">ℤ</span>-classes are not known for the given dimension, i.e., if <var class="Arg">dim</var> <span class="SimpleMath">> 11</span> and <var class="Arg">dim</var> <span class="SimpleMath">not ∈ { 13, 17, 19, 23 }</span>. The default value of <var class="Arg">z</var> is 1. This value of <var class="Arg">z</var> will be accepted even if the <span class="SimpleMath">ℤ</span>-classes are not known. Then it specifies the only representative group which is available for the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class. The greatest legal value of <var class="Arg">dim</var> is 31.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayImfInvariants( 3, 1, 0 );</span>
#I Z-class 3.1.1: Solvable, size = 2^4*3
#I isomorphism type = C2 wr S3 = C2 x S4 = W(B3)
#I elementary divisors = 1^3
#I orbit size = 6, minimal norm = 1
#I Z-class 3.1.2: Solvable, size = 2^4*3
#I isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)
#I elementary divisors = 1*4^2
#I orbit size = 8, minimal norm = 3
#I Z-class 3.1.3: Solvable, size = 2^4*3
#I isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)
#I elementary divisors = 1^2*4
#I orbit size = 12, minimal norm = 2
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayImfInvariants( 8, 15, 1 );</span>
#I Z-class 8.15.1: Solvable, size = 2^5*3^4
#I isomorphism type = C2 x (S3 wr S3)
#I elementary divisors = 1*3^3*9^3*27
#I orbit size = 54, minimal norm = 8
#I not maximal finite in GL(8,Q), rational imf class is 8.5
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayImfInvariants( 20, 23 );</span>
#I Q-class 20.23: Size = 2^5*3^2*5*11
#I isomorphism type = (PSL(2,11) x D12).C2
#I elementary divisors = 1^18*11^2
#I orbit size = 3*660 + 2*1980 + 2640 + 3960, minimal norm = 4
</pre></div>
<p>Note that the function <code class="func">DisplayImfInvariants</code> uses a kind of shorthand to display the elementary divisors. E. g., the expression <code class="code">1*3^3*9^3*27</code> in the preceding example stands for the elementary divisors <span class="SimpleMath">1,3,3,3,9,9,9,27</span>. (See also the next example which shows that the function <code class="func">ImfInvariants</code> (<a href="chap50.html#X8604A2167B2E8434"><span class="RefLink">50.12-3</span></a>) provides the elementary divisors in form of an ordinary <strong class="pkg">GAP</strong> list.)</p>
<p>In the description of the isomorphism types the following notations are used:</p>
<dl>
<dt><strong class="Mark"><span class="SimpleMath">A</span> <code class="code">x</code> <span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes a direct product of a group <span class="SimpleMath">A</span> by a group <span class="SimpleMath">B</span>,</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">A</span> <code class="code">subd</code> <span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes a subdirect product of <span class="SimpleMath">A</span> by <span class="SimpleMath">B</span>,</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">A</span> <code class="code">Y</code> <span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes a central product of <span class="SimpleMath">A</span> by <span class="SimpleMath">B</span>,</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">A</span> <code class="code">wr</code> <span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes a wreath product of <span class="SimpleMath">A</span> by <span class="SimpleMath">B</span>,</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">A</span><code class="code">:</code><span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes a split extension of <span class="SimpleMath">A</span> by <span class="SimpleMath">B</span>,</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">A</span><code class="code">.</code><span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes just an extension of <span class="SimpleMath">A</span> by <span class="SimpleMath">B</span> (split or nonsplit).</p>
</dd>
</dl>
<p>The groups involved are</p>
<ul>
<li><p>the cyclic groups <span class="SimpleMath">C_n</span>, dihedral groups <span class="SimpleMath">D_n</span>, and generalized quaternion groups <span class="SimpleMath">Q_n</span> of order <span class="SimpleMath">n</span>, denoted by <code class="code">C</code><var class="Arg">n</var>, <code class="code">D</code><var class="Arg">n</var>, and <code class="code">Q</code><var class="Arg">n</var>, respectively,</p>
</li>
<li><p>the alternating groups <span class="SimpleMath">A_n</span> and symmetric groups <span class="SimpleMath">S_n</span> of degree <span class="SimpleMath">n</span>, denoted by <code class="code">A</code><var class="Arg">n</var> and <code class="code">S</code><var class="Arg">n</var>, respectively,</p>
</li>
<li><p>the linear groups <span class="SimpleMath">GL_n(q)</span>, <span class="SimpleMath">PGL_n(q)</span>, <span class="SimpleMath">SL_n(q)</span>, and <span class="SimpleMath">PSL_n(q)</span>, denoted by <code class="code">GL</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), <code class="code">PGL</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), <code class="code">SL</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), and <code class="code">PSL</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), respectively,</p>
</li>
<li><p>the unitary groups <span class="SimpleMath">SU_n(q)</span> and <span class="SimpleMath">PSU_n(q)</span>, denoted by <code class="code">SU</code>(<var class="Arg">n</var>,<var class="Arg">q</var>) and <code class="code">PSU</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), respectively,</p>
</li>
<li><p>the symplectic groups <span class="SimpleMath">Sp(n,q)</span> and <span class="SimpleMath">PSp(n,q)</span>, denoted by <code class="code">Sp</code>(<var class="Arg">n</var>,<var class="Arg">q</var>) and <code class="code">PSp</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), respectively,</p>
</li>
<li><p>the orthogonal groups <span class="SimpleMath">O_8^+(2)</span> and <span class="SimpleMath">PO_8^+(2)</span>, denoted by <code class="code">O+</code>(8,2) and <code class="code">PO+</code>(8,2), respectively,</p>
</li>
<li><p>the extraspecial groups <span class="SimpleMath">2_+^{1+8}</span>, <span class="SimpleMath">3_+^{1+2}</span>, <span class="SimpleMath">3_+^{1+4}</span>, and <span class="SimpleMath">5_+^{1+2}</span>, denoted by <code class="code">2+^(1+8)</code>, <code class="code">3+^(1+2)</code>, <code class="code">3+^(1+4)</code>, and <code class="code">5+^(1+2)</code>, respectively,</p>
</li>
<li><p>the Chevalley group <span class="SimpleMath">G_2(3)</span>, denoted by <code class="code">G2(3)</code>,</p>
</li>
<li><p>the twisted Chevalley group <span class="SimpleMath">^3D_4(2)</span>, denoted by <code class="code">3D4(2)</code>,</p>
</li>
<li><p>the Suzuki group <span class="SimpleMath">Sz(8)</span>, denoted by <code class="code">Sz(8)</code>,</p>
</li>
<li><p>the Weyl groups <span class="SimpleMath">W(A_n)</span>, <span class="SimpleMath">W(B_n)</span>, <span class="SimpleMath">W(D_n)</span>, <span class="SimpleMath">W(E_n)</span>, and <span class="SimpleMath">W(F_4)</span>, denoted by <code class="code">W(A<var class="Arg">n</var>)</code>, <code class="code">W(B<var class="Arg">n</var>)</code>, <code class="code">W(D<var class="Arg">n</var>)</code>, <code class="code">W(E<var class="Arg">n</var>)</code>, and <code class="code">W(F4)</code>, respectively,</p>
</li>
<li><p>the sporadic simple groups <span class="SimpleMath">Co_1</span>, <span class="SimpleMath">Co_2</span>, <span class="SimpleMath">Co_3</span>, <span class="SimpleMath">HS</span>, <span class="SimpleMath">J_2</span>, <span class="SimpleMath">M_12</span>, <span class="SimpleMath">M_22</span>, <span class="SimpleMath">M_23</span>, <span class="SimpleMath">M_24</span>, and <span class="SimpleMath">Mc</span>, denoted by <code class="code">Co1</code>, <code class="code">Co2</code>, <code class="code">Co3</code>, <code class="code">HS</code>, <code class="code">J2</code>, <code class="code">M12</code>, <code class="code">M22</code>, <code class="code">M23</code>, <code class="code">M24</code>, and <code class="code">Mc</code>, respectively,</p>
</li>
<li><p>a point stabilizer of index 11 in <span class="SimpleMath">M_11</span>, denoted by <code class="code">M10</code>.</p>
</li>
</ul>
<p>As mentioned above, the data assembled by the function <code class="func">DisplayImfInvariants</code> are "cheap data" in the sense that they can be provided by the library without loading any of its large matrix files or performing any matrix calculations. The following function allows you to get proper access to these cheap data instead of just displaying them.</p>
<p><a id="X8604A2167B2E8434" name="X8604A2167B2E8434"></a></p>
<h5>50.12-3 ImfInvariants</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImfInvariants</code>( <var class="Arg">dim</var>, <var class="Arg">q</var>[, <var class="Arg">z</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">ImfInvariants</code> returns a record which provides some <span class="SimpleMath">ℤ</span>-class invariants of the groups in the <var class="Arg">z</var>-th <span class="SimpleMath">ℤ</span>-class in the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class of i.m.f. integral matrix groups of dimension <var class="Arg">dim</var>. A value <var class="Arg">z</var> <span class="SimpleMath">≠ 1</span> must not be specified if the <span class="SimpleMath">ℤ</span>-classes are not known for the given dimension, i.e., if <var class="Arg">dim</var> <span class="SimpleMath">> 11</span> and <var class="Arg">dim</var> <span class="SimpleMath">not ∈ { 13, 17, 19, 23 }</span>. The default value of <var class="Arg">z</var> is 1. This value of <var class="Arg">z</var> will be accepted even if the <span class="SimpleMath">ℤ</span>-classes are not known. Then it specifies the only representative group which is available for the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class. The greatest legal value of <var class="Arg">dim</var> is 31.</p>
<p>The resulting record contains six or seven components:</p>
<dl>
<dt><strong class="Mark"><code class="code">size</code> </strong></dt>
<dd><p>the size of any representative group <var class="Arg">G</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">isSolvable</code> </strong></dt>
<dd><p>is <code class="keyw">true</code> if <var class="Arg">G</var> is solvable,</p>
</dd>
<dt><strong class="Mark"><code class="code">isomorphismType</code> </strong></dt>
<dd><p>a text string describing the isomorphism type of <var class="Arg">G</var> (in the same notation as used by the function <code class="code">DisplayImfInvariants</code> above),</p>
</dd>
<dt><strong class="Mark"><code class="code">elementaryDivisors</code> </strong></dt>
<dd><p>the elementary divisors of the associated Gram matrix <var class="Arg">F</var> (in the same format as the result of the function <code class="func">ElementaryDivisorsMat</code> (<a href="chap24.html#X7AC4D74F81908109"><span class="RefLink">24.9-1</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">minimalNorm</code> </strong></dt>
<dd><p>the norm of the associated short vectors,</p>
</dd>
<dt><strong class="Mark"><code class="code">sizesOrbitsShortVectors</code> </strong></dt>
<dd><p>the sizes of the orbits of short vectors under <var class="Arg">F</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">maximalQClass</code> </strong></dt>
<dd><p>the <span class="SimpleMath">ℚ</span>-class number of an i.m.f. group in <span class="SimpleMath">GL_n(ℚ)</span> that contains <var class="Arg">G</var> as a subgroup (only in case that not <var class="Arg">G</var> itself is an i.m.f. subgroup of <span class="SimpleMath">GL_n(ℚ)</span>).</p>
</dd>
</dl>
<p>Note that four of these data, namely the group size, the solvability, the isomorphism type, and the corresponding rational i.m.f. class, are not only <span class="SimpleMath">ℤ</span>-class invariants, but also <span class="SimpleMath">ℚ</span>-class invariants.</p>
<p>Note further that, though the isomorphism type is a <span class="SimpleMath">ℚ</span>-class invariant, you will sometimes get different descriptions for different <span class="SimpleMath">ℤ</span>-classes of the same <span class="SimpleMath">ℚ</span>-class (as, e.g., for the classes 3.1.1 and 3.1.2 in the last example above). The purpose of this behaviour is to provide some more information about the underlying lattices.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImfInvariants( 8, 15, 1 );</span>
rec( elementaryDivisors := [ 1, 3, 3, 3, 9, 9, 9, 27 ],
isSolvable := true, isomorphismType := "C2 x (S3 wr S3)",
maximalQClass := 5, minimalNorm := 8, size := 2592,
sizesOrbitsShortVectors := [ 54 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ImfInvariants( 24, 1 ).size;</span>
10409396852733332453861621760000
<span class="GAPprompt">gap></span> <span class="GAPinput">ImfInvariants( 23, 5, 2 ).sizesOrbitsShortVectors;</span>
[ 552, 53130 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [ 1 .. ImfNumberQClasses( 22 ) ] do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Print( ImfInvariants( 22, i ).isomorphismType, "\n" ); od;</span>
C2 wr S22 = W(B22)
(C2 x PSU(6,2)).S3
(C2 x S3) wr S11 = (C2 x W(A2)) wr S11
(C2 x S12) wr C2 = (C2 x W(A11)) wr C2
C2 x S3 x S12 = C2 x W(A2) x W(A11)
(C2 x HS).C2
(C2 x Mc).C2
C2 x S23 = C2 x W(A22)
C2 x PSL(2,23)
C2 x PSL(2,23)
C2 x PGL(2,23)
C2 x PGL(2,23)
</pre></div>
<p><a id="X78935B307B909101" name="X78935B307B909101"></a></p>
<h5>50.12-4 ImfMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImfMatrixGroup</code>( <var class="Arg">dim</var>, <var class="Arg">q</var>[, <var class="Arg">z</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">ImfMatrixGroup</code> is the essential extraction function of this library (note that its name has been changed from <code class="code">ImfMatGroup</code> in <strong class="pkg">GAP</strong> 3 to <code class="func">ImfMatrixGroup</code> in <strong class="pkg">GAP</strong> 4). It returns a representative group, <span class="SimpleMath">G</span> say, of the <var class="Arg">z</var>-th <span class="SimpleMath">ℤ</span>-class in the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class of i.m.f. integral matrix groups of dimension <var class="Arg">dim</var>. A value <var class="Arg">z</var><span class="SimpleMath">≠ 1</span> must not be specified if the <span class="SimpleMath">ℤ</span>-classes are not known for the given dimension, i.e., if <var class="Arg">dim</var> <span class="SimpleMath">> 11</span> and <var class="Arg">dim</var> <span class="SimpleMath">not ∈ { 13, 17, 19, 23 }</span>. The default value of <var class="Arg">z</var> is 1. This value of <var class="Arg">z</var> will be accepted even if the <span class="SimpleMath">ℤ</span>-classes are not known. Then it specifies the only representative group which is available for the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class. The greatest legal value of <var class="Arg">dim</var> is 31.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ImfMatrixGroup( 5, 1, 3 );</span>
ImfMatrixGroup(5,1,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">for m in GeneratorsOfGroup( G ) do PrintArray( m ); od;</span>
[ [ -1, 0, 0, 0, 0 ],
[ 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 1, 0 ],
[ -1, -1, -1, -1, 2 ],
[ -1, 0, 0, 0, 1 ] ]
[ [ 0, 1, 0, 0, 0 ],
[ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ],
[ 1, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 1 ] ]
</pre></div>
<p>The attributes <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>) and <code class="code">IsSolvable</code> will be properly set in the resulting matrix group <span class="SimpleMath">G</span>. In addition, it has two attributes <code class="code">IsImfMatrixGroup</code> and <code class="code">ImfRecord</code> where the first one is just a logical flag set to <code class="keyw">true</code> and the latter one is a record. Except for the group size and the solvability flag, this record contains the same components as the resulting record of the function <code class="func">ImfInvariants</code> (<a href="chap50.html#X8604A2167B2E8434"><span class="RefLink">50.12-3</span></a>) described above, namely the components <code class="code">isomorphismType</code>, <code class="code">elementaryDivisors</code>, <code class="code">minimalNorm</code>, and <code class="code">sizesOrbitsShortVectors</code> and, if <span class="SimpleMath">G</span> is not a rational i.m.f. group, <code class="code">maximalQClass</code>. Moreover, it has the two components</p>
<dl>
<dt><strong class="Mark"><code class="code">form</code></strong></dt>
<dd><p>the associated Gram matrix <span class="SimpleMath">F</span>, and</p>
</dd>
<dt><strong class="Mark"><code class="code">repsOrbitsShortVectors</code></strong></dt>
<dd><p>representatives of the orbits of short vectors under <span class="SimpleMath">F</span>.</p>
</dd>
</dl>
<p>The last one of these components will be required by the function <code class="func">IsomorphismPermGroup</code> (<a href="chap50.html#X84BF34B27CD5E85C"><span class="RefLink">50.12-5</span></a>) below.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( G );</span>
3840
<span class="GAPprompt">gap></span> <span class="GAPinput">imf := ImfRecord( G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">imf.isomorphismType;</span>
"C2 wr S5 = C2 x W(D5)"
<span class="GAPprompt">gap></span> <span class="GAPinput">PrintArray( imf.form );</span>
[ [ 4, 0, 0, 0, 2 ],
[ 0, 4, 0, 0, 2 ],
[ 0, 0, 4, 0, 2 ],
[ 0, 0, 0, 4, 2 ],
[ 2, 2, 2, 2, 5 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">imf.elementaryDivisors;</span>
[ 1, 4, 4, 4, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">imf.minimalNorm;</span>
4
</pre></div>
<p>If you want to perform calculations in such a matrix group <span class="SimpleMath">G</span> you should be aware of the fact that the permutation group routines of <strong class="pkg">GAP</strong> are much more efficient than the matrix group routines. Hence we recommend that you do your computations, whenever possible, in the isomorphic permutation group which is induced by the action of <span class="SimpleMath">G</span> on one of the orbits of the associated short vectors. You may call one of the following functions <code class="func">IsomorphismPermGroup</code> (<a href="chap50.html#X84BF34B27CD5E85C"><span class="RefLink">50.12-5</span></a>) or <code class="func">IsomorphismPermGroupImfGroup</code> (<a href="chap50.html#X7CEDB6CE7BAC4518"><span class="RefLink">50.12-6</span></a>) to get an isomorphism to such a permutation group (note that these <strong class="pkg">GAP</strong> 4 functions have replaced the <strong class="pkg">GAP</strong> 3 functions <code class="code">PermGroup</code> and <code class="code">PermGroupImfGroup</code>).</p>
<p><a id="X84BF34B27CD5E85C" name="X84BF34B27CD5E85C"></a></p>
<h5>50.12-5 IsomorphismPermGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPermGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns an isomorphism, <span class="SimpleMath">φ</span> say, from the given i.m.f. integral matrix group <span class="SimpleMath">G</span> to a permutation group <span class="SimpleMath">P := φ(G)</span> acting on a minimal orbit, <span class="SimpleMath">S</span> say, of short vectors of <span class="SimpleMath">G</span> such that each matrix <span class="SimpleMath">m ∈ G</span> is mapped to the permutation induced by its action on <span class="SimpleMath">S</span>.</p>
<p>Note that in case of a large orbit the construction of <span class="SimpleMath">φ</span> may be space and time consuming. Fortunately, there are only six <span class="SimpleMath">ℚ</span>-classes in the library for which the smallest orbit of short vectors is of size greater than <span class="SimpleMath">20000</span>, the worst case being the orbit of size <span class="SimpleMath">196560</span> for the Leech lattice (<var class="Arg">dim</var> <span class="SimpleMath">= 24</span>, <var class="Arg">q</var> <span class="SimpleMath">= 3</span>).</p>
<p>The inverse isomorphism <span class="SimpleMath">φ^{-1}</span> from <span class="SimpleMath">P</span> to <span class="SimpleMath">G</span> is constructed by determining a <span class="SimpleMath">ℚ</span>-base <span class="SimpleMath">B ⊂ S</span> of <span class="SimpleMath">ℚ^{1 × dim}</span> in <span class="SimpleMath">S</span> and, in addition, the associated base change matrix <span class="SimpleMath">M</span> which transforms <span class="SimpleMath">B</span> into the standard base of <span class="SimpleMath">ℤ^{1 × dim}</span>. This allows a simple computation of the preimage <span class="SimpleMath">φ^{-1}(p)</span> of any permutation <span class="SimpleMath">p ∈ P</span>, as follows. If, for <span class="SimpleMath">1 ≤ i ≤</span> <var class="Arg">dim</var>, <span class="SimpleMath">b_i</span> is the position number in <span class="SimpleMath">S</span> of the <span class="SimpleMath">i</span>-th base vector in <span class="SimpleMath">B</span>, it suffices to look up the vector whose position number in <span class="SimpleMath">S</span> is the image of <span class="SimpleMath">b_i</span> under <span class="SimpleMath">p</span> and to multiply this vector by <span class="SimpleMath">M</span> to get the <span class="SimpleMath">i</span>-th row of <span class="SimpleMath">φ^{-1}(p)</span>.</p>
<p>You may use the functions <code class="func">Image</code> (<a href="chap32.html#X87F4D35A826599C6"><span class="RefLink">32.4-6</span></a>) and <code class="func">PreImage</code> (<a href="chap32.html#X836FAEAC78B55BF4"><span class="RefLink">32.5-6</span></a>) to switch from <span class="SimpleMath">G</span> to <span class="SimpleMath">P</span> and back from <span class="SimpleMath">P</span> to <span class="SimpleMath">G</span>.</p>
<p>As an example, let us continue the preceding example and compute the solvable residuum of the group <span class="SimpleMath">G</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># Perform the computations in an isomorphic permutation group.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := IsomorphismPermGroup( G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P := Image( phi );</span>
Group([ (1,7,6)(2,9)(4,5,10), (2,3,4,5)(6,9,8,7) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">D := DerivedSubgroup( P );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( D );</span>
960
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPerfectGroup( D );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput"># We have found the solvable residuum of P,</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># now move the results back to the matrix group G.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := PreImage( phi, D );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(R);</span>
"(C2 x C2 x C2 x C2) : A5"
<span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup(D)=IdGroup(R);</span>
true
</pre></div>
<p><a id="X7CEDB6CE7BAC4518" name="X7CEDB6CE7BAC4518"></a></p>
<h5>50.12-6 IsomorphismPermGroupImfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPermGroupImfGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">IsomorphismPermGroupImfGroup</code> returns an isomorphism, <span class="SimpleMath">φ</span> say, from the given i.m.f. integral matrix group <var class="Arg">G</var> to a permutation group <span class="SimpleMath">P</span> acting on the <var class="Arg">n</var>-th orbit, <span class="SimpleMath">S</span> say, of short vectors of <var class="Arg">G</var> such that each matrix <span class="SimpleMath">m ∈</span> <var class="Arg">G</var> is mapped to the permutation induced by its action on <span class="SimpleMath">S</span>.</p>
<p>The only difference to the above function <code class="func">IsomorphismPermGroup</code> (<a href="chap50.html#X84BF34B27CD5E85C"><span class="RefLink">50.12-5</span></a>) is that you can specify the orbit to be used. In fact, as the orbits of short vectors are sorted by increasing sizes, the function <code class="code">IsomorphismPermGroup( <var class="Arg">G</var> )</code> has been implemented such that it is equivalent to <code class="code">IsomorphismPermGroupImfGroup( <var class="Arg">G</var>, 1 )</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImfInvariants( 12, 9 ).sizesOrbitsShortVectors;</span>
[ 120, 300 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ImfMatrixGroup( 12, 9 );</span>
ImfMatrixGroup(12,9)
<span class="GAPprompt">gap></span> <span class="GAPinput">phi1 := IsomorphismPermGroupImfGroup( G, 1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P1 := Image( phi1 );</span>
<permutation group of size 2400 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestMovedPoint( P1 );</span>
120
<span class="GAPprompt">gap></span> <span class="GAPinput">phi2 := IsomorphismPermGroupImfGroup( G, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P2 := Image( phi2 );</span>
<permutation group of size 2400 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestMovedPoint( P2 );</span>
300
</pre></div>
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