gap-libs 4r7p5-2 / usr / share / gap / lib / ctblsolv.gi

#############################################################################
##
#W  ctblsolv.gi                 GAP library                Hans Ulrich Besche
#W                                                              Thomas Breuer
##
##
#Y  Copyright (C)  1997,  Lehrstuhl D für Mathematik,  RWTH Aachen,  Germany
#Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y  Copyright (C) 2002 The GAP Group
##
##  This file contains character table methods for solvable groups.
##


#############################################################################
##
#M  CharacterDegrees( <G>, <p> )  . . . . . . . . . . .  for an abelian group
##
InstallMethod( CharacterDegrees,
    "for an abelian group, and an integer p (just strip off the p-part)",
    [ IsGroup and IsAbelian, IsInt ],
    RankFilter(IsZeroCyc), # There is a method for groups for
                           # the integer zero which is worse
    function( G, p )
    G:= Size( G );
    if p <> 0 then
      while G mod p = 0 do
        G:= G / p;
      od;
    fi;
    return [ [ 1, G ] ];
    end );


#############################################################################
##
#F  AppendCollectedList( <list1>, <list2> )
##
BindGlobal( "AppendCollectedList", function( list1, list2 )
    local pair1, pair2, toadd;
    for pair2 in list2 do
      toadd:= true;
      for pair1 in list1 do
        if pair1[1] = pair2[1] then
          pair1[2]:= pair1[2] + pair2[2];
          toadd:= false;
          break;
        fi;
      od;
      if toadd then
        AddSet( list1, pair2 );
      fi;
    od;
end );


#############################################################################
##
#F  KernelUnderDualAction( <N>, <Npcgs>, <v> )  . . . . . . .  local function
##
##  <Npcgs> is a PCGS of an elementary abelian group <N>.
##  <v> is a vector in the dual space of <N>, w.r.t. <Npcgs>.
##  The kernel of <v> is returned.
##
BindGlobal( "KernelUnderDualAction", function( N, Npcgs, v )
    local gens, # generators list
          i, j;

    gens:= [];
    for i in Reversed( [ 1 .. Length( v ) ] ) do
      if IsZero( v[i] ) then
        Add( gens, Npcgs[i] );
      else
        # `i' is the position of the last nonzero entry of `v'.
        for j in Reversed( [ 1 .. i-1 ] ) do
          Add( gens, Npcgs[j]*Npcgs[i]^( Int(-v[j]/v[i]) ) );
        od;
        return SubgroupNC( N, Reversed( gens ) );
      fi;
    od;
end );


#############################################################################
##
#F  ProjectiveCharDeg( <G> ,<z> ,<q> )
##
InstallGlobalFunction( ProjectiveCharDeg, function( G, z, q )
    local oz,       # the order of `z'
          N,        # normal subgroup of `G'
          t,
          r,        # collected list of character degrees, result
          h,        # natural homomorphism
          img,
          k,
          c,
          ci,
          zn,
          i,
          p,        # prime divisor of the size of `N'
          P,        # Sylow `p' subgroup of `N'
          O,
          L,
          Gpcgs,    # PCGS of `G'
          Ppcgs,    # PCGS of `P'
          Opcgs,    # PCGS of `O'
          mats,
          orbs,
          orb,      # loop over `orbs'
          stab;     # stabilizer of canonical representative of `orb'

    oz:= Order( z );

    # For abelian groups, there are only linear characters.
    if IsAbelian( G ) then
      G:= Size( G );
      if q <> 0 then
        while G mod q = 0 do
          G:= G / q;
        od;
      fi;
      return [ [ 1, G/oz ] ];
    fi;

    # Now `G' is not abelian.
    h:= NaturalHomomorphismByNormalSubgroupNC( G, SubgroupNC( G, [ z ] ) );
    img:= ImagesSource( h );
    N:= ElementaryAbelianSeriesLargeSteps( img );
    N:= N[ Length( N )-1 ];
    if not IsPrime( Size( N ) ) then
      N:= ChiefSeriesUnderAction( img, N );
      N:= N[ Length( N )-1 ];
    fi;

    # `N' is a normal subgroup such that `N/<z>' is a chief factor of `G'
    # of order `i' which is a power of `p'.
    N:= PreImagesSet( h, N );
    i:= Size( N ) / oz;
    p:= Factors( i )[1];

    if not IsAbelian( N ) then

      h:= NaturalHomomorphismByNormalSubgroupNC( G, SubgroupNC( G, [ z ] ) );

      # `c' is a list of complement classes of `N' modulo `z'
      c:= List( ComplementClassesRepresentatives( ImagesSource( h ), ImagesSet( h, N ) ),
                x -> PreImagesSet( h, x ) );
      r:= Centralizer( G, N );
      for L in c do
        if IsSubset( L, r ) then

          # L is a complement to N in G modulo <z> which centralizes N
          r:= RootInt( Size(N) / oz );
          return List( ProjectiveCharDeg( L, z, q ),
                       x -> [ x[1]*r, x[2] ] );

        fi;
      od;
      Error( "this should not happen" );

    fi;

    # `N' is abelian, `P' is its Sylow `p' subgroup.
    P:= SylowSubgroup( N, p );

    if p = q then

      # Factor out `P' (lies in the kernel of the repr.)
      h:= NaturalHomomorphismByNormalSubgroupNC( G, P );
      return ProjectiveCharDeg( ImagesSource( h ), ImageElm( h, z ), q );

    elif i = Size( P ) then

      # `z' is a p'-element, `P' is elementary abelian.
      # Find the characters of the factor group needed.
      h:= NaturalHomomorphismByNormalSubgroupNC( G, P );
      r:= ProjectiveCharDeg( ImagesSource( h ), ImageElm( h, z ), q );

      if p = i then

        # `P' has order `p'.
        zn:= First( GeneratorsOfGroup( P ), g -> not IsOne( g ) );
        t:=  Stabilizer( G, zn );
        i:= Size(G) / Size(t);
        AppendCollectedList( r,
            List( ProjectiveCharDeg( t, zn*z, q ),
                  x -> [ x[1]*i, x[2]*(p-1)/i ] ) );
        return r;

      else

        # `P' has order strictly larger than `p'.
        # `mats' describes the contragredient operation of `G' on `P'.
        Gpcgs:= Pcgs( G );
        Ppcgs:= Pcgs( P );
        mats:= List( List( Gpcgs, Inverse ),
                   x -> TransposedMat( List( Ppcgs,
                   y -> ExponentsConjugateLayer( Ppcgs, y,x ) )*Z(p)^0 ) );
        orbs:= ExternalOrbitsStabilizers( G,
                   NormedRowVectors( GF(p)^Length( Ppcgs ) ),
                   Gpcgs, mats, OnLines );
        orbs:= Filtered( orbs,
              o -> not IsZero( CanonicalRepresentativeOfExternalSet( o ) ) );

        for orb in orbs do

          # `k' is the kernel of the character.
          stab:= StabilizerOfExternalSet( orb );
          h:= NaturalHomomorphismByNormalSubgroupNC( stab,
                  KernelUnderDualAction( P, Ppcgs,
                      CanonicalRepresentativeOfExternalSet( orb ) ) );
          img:= ImagesSource( h );

          # `zn' is an element of `img'.
          # Note that the image of `P' under `h' has order `p'.
          zn:= First( GeneratorsOfGroup( ImagesSet( h, P) ),
                      g -> not IsOne( g ) )
               * ImageElm( h, z );

          # `c' is stabilizer of the character,
          # `ci' is the number of orbits of characters with equal kernels
          if p = 2 then
            c  := img;
            ci := 1;
          else
            c  := Stabilizer( img, zn );
            ci := Size( img ) / Size( c );
          fi;
          k:= Size( G ) / Size( stab ) * ci;
          AppendCollectedList( r,
              List( ProjectiveCharDeg( c, zn, q ),
                    x -> [ x[1]*k, x[2]*(p-1)/ci ] ) );

        od;
        return r;

      fi;

    elif IsCyclic( P ) then

      # Choose a generator `zn' of `P'.
      zn := Pcgs( P )[1];
      t  := Stabilizer( G, zn, OnPoints );
      if G = t then
        # `P' is a central subgroup of `G'.
        return List( ProjectiveCharDeg( G, zn*z, q ),
                     x -> [ x[1], x[2]*p ] );
      else
        # `P' is not central in `G'.
        return List( ProjectiveCharDeg( t, zn*z, q ),
                     x -> [ x[1]*p, x[2] ] );
      fi;

    fi;

    # `P' is the direct product of the Sylow `p' subgroup of `z'
    # and an elementary abelian `p' subgroup.
    O:= Omega( P, p );
    Opcgs:= Pcgs( O );
    Gpcgs:= Pcgs( G );

    # `zn' is a generator of the intersection of <z> and `O'
    zn := z^(oz/p);
    r  := [];
    mats:= List( List( Gpcgs, Inverse ),
                 x -> TransposedMat( List( Opcgs,
                      y -> ExponentsConjugateLayer( Opcgs, y,x ) ) * Z(p)^0 ) );
    orbs:= ExternalOrbitsStabilizers( G,
               NormedRowVectors( GF(p)^Length( Opcgs ) ),
               Gpcgs, mats, OnLines );
    orbs:= Filtered( orbs,
              o -> not IsZero( CanonicalRepresentativeOfExternalSet( o ) ) );

    # In this case the stabilzers of the kernels are already the
    # stabilizers of the characters.
    for orb in orbs do
      k:= KernelUnderDualAction( O, Opcgs,
              CanonicalRepresentativeOfExternalSet( orb ) );
      if not zn in k then
        # The kernel avoids `zn'.
        t:= StabilizerOfExternalSet( orb );
        h:= NaturalHomomorphismByNormalSubgroupNC( t, k );
        img:= ImagesSource( h );
        t:= Size(G) / Size(t);
        AppendCollectedList( r, List( ProjectiveCharDeg( img,
                                          ImageElm( h, z ), q ),
                                      x -> [ x[1]*t, x[2] ] ) );
      fi;
    od;
    return r;
end );


#############################################################################
##
#M  CharacterDegrees( <G>, <p> )  . . . . . . . . . . .  for a solvable group
##
##  The algorithm used is based on~\cite{Con90b},
##  its main tool is Clifford theory.
##
##  Given a solvable group $G$ and a nonnegative integer $q$,
##  we first choose an elementary abelian normal subgroup $N$.
##  (Note that $N$ need not be a *minimal* normal subgroup, this requirement
##  in~\cite{Con90b} applies only to the computation of projective degrees
##  where nonabelian normal subgroups $N$ occur.)
##  By recursion, the $q$-modular character degrees of the factor group $G/N$
##  are computed next.
##  So it remains to compute the degrees of those $q$-modular irreducible
##  characters whose kernels do not contain $N$.
##  This last step follows~\cite{Con90b}, for the special case of a *trivial*
##  central subgroup $Z$.
##  Namely, we compute the $G$-orbits on the linear spaces of the nontrivial
##  irreducible characters of $N$, under projective action.
##  (The orbit consisting of the trivial character corresponds to those
##  $q$-modular irreducible $G$-characters with $N$ in their kernels.)
##  For each orbit, we use the function `ProjectiveCharDeg' to compute the
##  degrees arising from a representative $\chi$,
##  in the group $S/K$ with central cyclic subgroup $N/K$,
##  where $S$ is the (subspace) stabilizer of $\chi$ and $K$ is the kernel of
##  $\chi$.
##
##  One recursive step of the algorithm is described in the following.
##
##  Let $G$ be a solvable group, $z$ a central element in $G$,
##  and let $q$ be the characteristic of the algebraic closed field $F$.
##  Without loss of generality, we may assume that $G$ is nonabelian.
##  Consider a faithful linear character $\lambda$ of $\langle z \rangle$.
##  We calculate the character degrees $(G,z,q)$ of those absolutely
##  irreducible characters of $G$ whose restrictions to $\langle z \rangle$
##  are a multiple of $\lambda$.
##
##  We choose a normal subgroup $N$ of $G$ such that the factor
##  $N / \langle z \rangle$ is a chief factor in $G$, and consider
##  the following cases.
##
##  If $N$ is nonabelian then we calculate a subgroup $L$ of $G$ such that
##  $N \cap L = \langle z \rangle$, $L$ centralizes $N$, and $N L = G$.
##  One can show that the order of $N / \langle z \rangle$ is a square $r^2$,
##  and that the degrees $(G,z,q)$ are obtained from the degrees $(L,z,q)$
##  on multiplying each with $r$.
##
##  If $N$ is abelian then the order of $N / \langle z \rangle$ is a prime
##  power $p^i$.
##  Let $P$ denote the Sylow $p$ subgroup of $N$.
##  Following Clifford's theorem, we calculate orbit representatives and
##  inertia subgroups with respect to the action of $G$ on those irreducible
##  characters of $P$ that restrict to multiples of $\lambda_P$.
##  For that, we distinguish three cases.
##  \beginlist
##  \item{(a)}
##      $z$ is a $p^{\prime}$ element.
##      Then we compute first the character degrees $(G/P,zP,q)$,
##      corresponding to the (orbit of the) trivial character.
##      The action on the nontrivial irreducible characters of $P$
##      is dual to the action on the nonzero vectors of the vector space
##      $P$.
##      For each representative, we compute the kernel $K$, and the degrees
##      $(S/K,zK,q)$, where $S$ denotes the inertia subgroup.
##
##  \item{(b)}
##      $z$ is not a $p^{\prime}$ element, and $P$ cyclic (not prime order).
##      Let $y$ be a generator of $P$.
##      If $y$ is central in $G$ then we have to return $p$ copies of the
##      degrees $(G,zy,q)$.
##      Otherwise we compute the degrees $(C_G(y),zy,q)$, and multiply
##      each with $p$.
##
##  \item{(c)}
##      $z$ is not a $p^{\prime}$ element, and $P$ is not cyclic.
##      We compute $O = \Omega(P)$.
##      As above, we consider the dual operation to that in $O$,
##      and for each orbit representative we check whether its restriction
##      to $O$ is a multiple of $\lambda_O$, and if yes compute the degrees
##      $(S/K,zK,q)$.
##  \endlist
##
BindGlobal( "CharacterDegreesConlon", function( G, q )
    local r,      # list of degrees, result
          N,      # elementary abelian normal subgroup of `G'
          p,      # prime divisor of the order of `N'
          z,      # one generator of `N'
          t,      # stabilizer of `z' in `G'
          i,      # index of `t' in `G'
          Gpcgs,  # PCGS of `G'
          Npcgs,  # PCGS of `N'
          mats,   # matrices describing the action of `Gpcgs' w.r.t. `Npcgs'
          orbs,   # orbits of the action
          orb,    # loop over `orbs'
          rep,    # canonical representative of `orb'
          stab,   # stabilizer of `rep'
          h,      # nat. hom. by the kernel of a character
          img,    # image of `h'
          c,
          ci,
          k;

    Info( InfoCharacterTable, 1,
          "CharacterDegrees: called for group of order ", Size( G ) );

    # If the group is abelian, we must give up because this method
    # needs a proper elementary abelian normal subgroup for its
    # reduction step.
    # (Note that we must not call `TryNextMethod' because the method
    # for abelian groups has higher rank.)
    if IsAbelian( G ) then
      r:= CharacterDegrees( G, q );
      Info( InfoCharacterTable, 1,
            "CharacterDegrees: returns ", r );
      return r;
    elif not ( q = 0 or IsPrimeInt( q ) ) then
      Error( "<q> mut be zero or a prime" );
    fi;

    # Choose a normal elementary abelian `p'-subgroup `N',
    # not necessarily minimal.
    N:= ElementaryAbelianSeriesLargeSteps( G );
    N:= N[ Length( N ) - 1 ];
    r:= CharacterDegrees( G / N, q );
    p:= Factors( Size( N ) )[1];

    if p = q then

      # If `N' is a `q'-group we are done.
      Info( InfoCharacterTable, 1,
            "CharacterDegrees: returns ", r );
      return r;

    elif Size( N ) = p then

      # `N' is of prime order.
      z:= Pcgs( N )[1];
      t:= Stabilizer( G, z, OnPoints );
      i:= Size( G ) / Size( t );
      AppendCollectedList( r, List( ProjectiveCharDeg( t, z, q ),
                                    x -> [ x[1]*i, x[2]*(p-1)/i ] ) );

    else

      # `N' is an elementary abelian `p'-group of nonprime order.
      Gpcgs:= Pcgs( G );
      Npcgs:= Pcgs( N );
      mats:= List( Gpcgs, x -> TransposedMat( List( Npcgs,
                 y -> ExponentsConjugateLayer( Npcgs, y,x ) ) * Z(p)^0 )^-1 );
      orbs:= ExternalOrbitsStabilizers( G,
                 NormedRowVectors( GF( p )^Length( Npcgs ) ),
                 Gpcgs, mats, OnLines );
#T may fail because the list is too long!
      orbs:= Filtered( orbs,
              o -> not IsZero( CanonicalRepresentativeOfExternalSet( o ) ) );

      for orb in orbs do

        stab:= StabilizerOfExternalSet( orb );
        rep:= CanonicalRepresentativeOfExternalSet( orb );
        h:= NaturalHomomorphismByNormalSubgroupNC( stab,
                KernelUnderDualAction( N, Npcgs, rep ) );
        img:= ImagesSource( h );

        # The kernel has index `p' in `stab'.
        z:= First( GeneratorsOfGroup( ImagesSet( h, N ) ),
                   g -> not IsOne( g ) );
        if p = 2 then
          c  := img;
          ci := 1;
        else
          c  := Stabilizer( img, z );
          ci := Size( img ) / Size( c );
        fi;
        k:= Size( G ) / Size( stab ) * ci;
        AppendCollectedList( r, List( ProjectiveCharDeg( c, z, q ),
                                      x -> [ x[1]*k, x[2]*(p-1)/ci ] ) );

      od;

    fi;

    Info( InfoCharacterTable, 1,
          "CharacterDegrees: returns ", r );
    return r;
    end );

InstallMethod( CharacterDegrees,
    "for a solvable group and an integer (Conlon's algorithm)",
    [ IsGroup and IsSolvableGroup, IsInt ],
    RankFilter(IsZeroCyc), # There is a method for groups for
                           # the integer zero which is worse
    function( G, q )
    if HasIrr( G ) then
      # Use the known irreducibles.
      TryNextMethod();
    else
      return CharacterDegreesConlon( G, q );
    fi;
    end );


#############################################################################
##
#F  CoveringTriplesCharacters( <G>, <z> ) . . . . . . . . . . . . . . . local
##
InstallGlobalFunction( CoveringTriplesCharacters, function( G, z )
    local oz,
          h,
          img,
          N,
          t,
          r,
          k,
          c,
          zn,
          i,
          p,
          P,
          O,
          Gpcgs,
          Ppcgs,
          Opcgs,
          mats,
          orbs,
          orb;

    # The trivial character will be dealt with separately.
    if IsTrivial( G ) then
      return [];
    fi;

    oz:= Order( z );
    if Size( G ) = oz then
      return [ [ G, TrivialSubgroup( G ), z ] ];
    fi;

    h:= NaturalHomomorphismByNormalSubgroupNC( G, SubgroupNC( G, [ z ] ) );
    img:= ImagesSource( h );
    N:= ElementaryAbelianSeriesLargeSteps( img );
    N:= N[ Length( N ) - 1 ];
    if not IsPrime( Size( N ) ) then
      N:= ChiefSeriesUnderAction( img, N );
      N:= N[ Length( N ) - 1 ];
    fi;
    N:= PreImagesSet( h, N );

    if not IsAbelian( N ) then
      Info( InfoCharacterTable, 2,
            "#I misuse of `CoveringTriplesCharacters'!\n" );
      return [];
    fi;

    i:= Size( N ) / oz;
    p:= Factors( i )[1];
    P:= SylowSubgroup( N, p );

    if i = Size( P ) then

      # `z' is a p'-element, `P' is elementary abelian.
      # Find the characters of the factor group needed.
      h:= NaturalHomomorphismByNormalSubgroupNC( G, P );
      r:= List( CoveringTriplesCharacters( ImagesSource( h ),
                                           ImageElm( h, z ) ),
                x -> [ PreImagesSet( h, x[1] ),
                       PreImagesSet( h, x[2] ),
                       PreImagesRepresentative( h, x[3] ) ] );

      if p = i then

        # `P' has order `p'.
        zn:= First( GeneratorsOfGroup( P ), g -> not IsOne( g ) );
        return Concatenation( r,
                   CoveringTriplesCharacters( Stabilizer( G, zn ), zn*z ) );

      else

        Gpcgs:= Pcgs( G );
        Ppcgs:= Pcgs( P );
        mats:= List( List( Gpcgs, Inverse ),
                   x -> TransposedMat( List( Ppcgs,
                   y -> ExponentsConjugateLayer( Ppcgs, y,x ) )*Z(p)^0 ) );
        orbs:= ExternalOrbitsStabilizers( G,
                   NormedRowVectors( GF(p)^Length( Ppcgs ) ),
                   Gpcgs, mats, OnLines );
        orbs:= Filtered( orbs,
              o -> not IsZero( CanonicalRepresentativeOfExternalSet( o ) ) );

        for orb in orbs do
          h:= NaturalHomomorphismByNormalSubgroupNC(
                  StabilizerOfExternalSet( orb ),
                  KernelUnderDualAction( P, Ppcgs,
                      CanonicalRepresentativeOfExternalSet( orb ) ) );
          img:= ImagesSource( h );
          zn:= First( GeneratorsOfGroup( ImagesSet( h, P ) ),
                      g -> not IsOne( g ) )
               * ImageElm( h, z );

          if p = 2 then
            c:= img;
          else
            c:= Stabilizer( img, zn );
          fi;
          Append( r, List( CoveringTriplesCharacters( c, zn ),
                           x -> [ PreImagesSet( h, x[1] ),
                                  PreImagesSet( h, x[2] ),
                                  PreImagesRepresentative( h, x[3] ) ] ) );
        od;
        return r;

      fi;

    elif IsCyclic( P ) then

      zn:= Pcgs( P )[1];
      return CoveringTriplesCharacters( Stabilizer( G, zn ), zn*z );

    fi;

    O:= Omega( P, p );
    Opcgs:= Pcgs( O );
    Gpcgs:= Pcgs( G );

    zn := z^(oz/p);
    r  := [];
    mats:= List( List( Gpcgs, Inverse ),
                 x -> TransposedMat( List( Opcgs,
                      y -> ExponentsConjugateLayer( Opcgs, y,x ) )*Z(p)^0 ) );
    orbs:= ExternalOrbitsStabilizers( G,
               NormedRowVectors( GF(p)^Length( Opcgs ) ),
               Gpcgs, mats, OnLines );
    orbs:= Filtered( orbs,
              o -> not IsZero( CanonicalRepresentativeOfExternalSet( o ) ) );

    for orb in orbs do
      k:= KernelUnderDualAction( O, Opcgs,
              CanonicalRepresentativeOfExternalSet( orb ) );
      if not zn in k then
        t:= SubgroupNC( G, StabilizerOfExternalSet( orb ) );
        h:= NaturalHomomorphismByNormalSubgroupNC( t, k );
        img:= ImagesSource( h );
        Append( r,
            List( CoveringTriplesCharacters( img, ImageElm( h, z ) ),
                  x -> [ PreImagesSet( h, x[1] ),
                         PreImagesSet( h, x[2] ),
                         PreImagesRepresentative( h, x[3] ) ] ) );
      fi;
    od;
    return r;
end );


#############################################################################
##
#M  IrrConlon( <G> )
##
##  This algorithm is a generalization of the algorithm to compute the
##  absolutely irreducible degrees of a solvable group to the computation
##  of the absolutely irreducible characters of a supersolvable group,
##  using an idea like in
##
##      S. B. Conlon, J. Symbolic Computation (1990) 9, 535-550.
##
##  The function `CoveringTriplesCharacters' is used to compute a list of
##  triples describing linear representations of subgroups of <G>.
##  These linear representations are induced to <G> and then evaluated on
##  representatives of the conjugacy classes.
##
##  For every irreducible character the monomiality information is stored as
##  value of the attribute `TestMonomial'.
##
InstallMethod( IrrConlon,
    "for a group",
    [ IsGroup ],
    function( G )
    local mulmoma,    # local function: multiply monomial matrices
          ct,         # character table of `G'
          ccl,        # conjugacy classes of `G'
          Gpcgs,      # PCGS of `G'
          irr,        # matrix of character values
          irredinfo,  # monomiality info
          evl,        # encode class representatives as words in `Gpcgs'
          i,
          t,
          chi,
          j,
          mat,
          k,
          triple,
          hom,
          zi,
          oz,
          ee,
          zp,
          co,         # cosets
          coreps,     # representatives of `co'
          dim,
          rep,        # matrix representation
          bco,
          p,
          i1,         # loop variable in `mulmoma'
          re;         # result of `mulmoma'

    # Compute the product of the monomial matrices `a' and `b';
    # The diagonal elements are powers of a fixed `oz'-th root of unity.
    mulmoma:= function( a, b )
      re:= rec( perm:= [], diag:= [] );
      for i1 in [ 1 .. Length( a.perm ) ] do
        re.perm[i1]:= b.perm[ a.perm[i1] ];
        re.diag[ b.perm[i1] ]:= ( b.diag[ b.perm[i1] ] + a.diag[i1] ) mod oz;
      od;
      return re;
    end;

    ct:= CharacterTable( G );
    ccl:= ConjugacyClasses( ct );
    Gpcgs:= Pcgs( G );
    irr:= [];
    irredinfo:= [ rec( inducedFrom:= rec( subgroup:= G, kernel:= G ) ) ];

    # `evl' is a list describing representatives of the nontrivial
    # conjugacy classes.
    # the entry for the element $g.1^2*g.2^0*g.3^1$ is $[ 1, 1, 3 ]$.
    evl:= [];
    for i in [ 2 .. Length( ccl ) ] do
      k:= ExponentsOfPcElement( Gpcgs, Representative( ccl[i] ) );
      t:= [];
      for j in [ 1 .. Length( k ) ] do
        if 0 < k[j] then
          Append( t, [ 1 .. k[j] ]*0 + j );
        fi;
      od;
      Add( evl, t );
    od;

    for triple in CoveringTriplesCharacters( G, One( G ) ) do

      hom:= NaturalHomomorphismByNormalSubgroupNC( triple[1], triple[2] );
      zi:= ImagesRepresentative( hom, triple[3] );
      oz:= Order( zi );
      ee:= E( oz );
      zp:= List( [ 1 .. oz ], x -> zi^x );
      co:= RightCosets( G, triple[1] );
      coreps:= List(  co, Representative );
      dim:= Length( co );

      # `rep' describes a matrix representation on a module with basis
      # a transversal of the stabilizer in `G'.
      # (The monomial matrices are the same as in `RepresentationsPGroup'.)
      rep:= [];
      for i in Gpcgs do
        mat:= rec( perm:= [], diag:= [] );
        for j in [ 1 .. dim ] do
          bco:= co[j]*i;
          p:= Position( co, bco, 0 );
          Add( mat.perm, p );
          mat.diag[p]:= Position( zp,
              ImageElm( hom, coreps[j]*i*Inverse( coreps[p] ) ), 0 );
        od;
        Add( rep, mat );
      od;

      # Compute the representing matrices for class representatives,
      # and their traces.
      chi:= [ dim ];
      for j in evl do
        mat:= Iterated( rep{ j }, mulmoma );
        t:= 0;
        for k in [ 1 .. dim ] do
          if mat.perm[k] = k then
            t:= t + ee^mat.diag[k];
          fi;
        od;
        Add( chi, t );
      od;

      # Test if `chi' is known and add `chi' and its Galois-conjugates
      # to the list.
      # Also compute the monomiality information.
      if not chi in irr then
        chi:= GaloisMat( [ chi ] ).mat;
        Append( irr, chi );
        for j in chi do
          Add( irredinfo, rec( subgroup:= triple[1], kernel:= triple[2] ) );
        od;
      fi;

    od;

    # Construct the characters from their values lists,
    # and set the monomiality info.
    irr:= Concatenation( [ TrivialCharacter( G ) ],
                         List( irr, chi -> Character( ct, chi ) ) );
    for i in [ 1 .. Length( irr ) ] do
      SetTestMonomial( irr[i], irredinfo[i] );
    od;

    # Return the characters.
    return irr;
    end );


#############################################################################
##
#M  Irr( <G>, 0 ) . . . . . .  for a supersolvable group (Conlon's algorithm)
##
InstallMethod( Irr,
    "for a supersolvable group (Conlon's algorithm)",
    [ IsGroup and IsSupersolvableGroup, IsZeroCyc ],
    function( G, zero )
    local irr;
    irr:= IrrConlon( G );
    SetIrr( OrdinaryCharacterTable( G ), irr );
    return irr;
    end );

InstallMethod( Irr,
    "for a supersolvable group with known `IrrConlon'",
    [ IsGroup and IsSupersolvableGroup and HasIrrConlon, IsZeroCyc ],
    function( G, zero )
    local irr;
    irr:= IrrConlon( G );
    SetIrr( OrdinaryCharacterTable( G ), irr );
    return irr;
    end );


#############################################################################
##
#M  Irr( <G>, 0 ) . . . .  for a supersolvable group (Baum-Clausen algorithm)
##
InstallMethod( Irr,
    "for a supersolvable group (Baum-Clausen algorithm)",
    [ IsGroup and IsSupersolvableGroup, IsZeroCyc ],
    function( G, zero )
    local irr;
    irr:= IrrBaumClausen( G );
    SetIrr( OrdinaryCharacterTable( G ), irr );
    return irr;
    end );

InstallMethod( Irr,
    "for a supersolvable group with known `IrrBaumClausen'",
    [ IsGroup and IsSupersolvableGroup and HasIrrBaumClausen, IsZeroCyc ],
    function( G, zero )
    local irr;
    irr:= IrrBaumClausen( G );
    SetIrr( OrdinaryCharacterTable( G ), irr );
    return irr;
    end );


#############################################################################
##
#V  BaumClausenInfoDebug  . . . . . . . . . . . . . . testing BaumClausenInfo
##
InstallValue( BaumClausenInfoDebug, rec(
    makemat:= function( record, e )
        local dim, mat, diag, gcd, i;
        dim:= Length( record.diag );
        mat:= NullMat( dim, dim );
        diag:= record.diag;
        gcd:= Gcd( diag );
        if gcd = 0 then
          e:= 1;
        else
          gcd:= GcdInt( gcd, e );
          e:= E( e / gcd );
          diag:= diag / gcd;
        fi;
        for i in [ 1 .. dim ] do
          mat[i][ record.perm[i] ]:= e^diag[ record.perm[i] ];
        od;
        return mat;
    end,

    testrep:= function( pcgs, rep, e )
        local images, hom;
        images:= List( rep,
                       record -> BaumClausenInfoDebug.makemat( record, e ) );
        hom:= GroupGeneralMappingByImagesNC( Group( pcgs ), Group( images ),
                                           pcgs, images );
        return IsGroupHomomorphism( hom );
    end,

    checkconj:= function( pcgs, i, lg, j, rep1, rep2, X, e )
        local ii, exps, mat, jj;
        X:= BaumClausenInfoDebug.makemat( X, e );
        for ii in [ i .. lg ] do
          exps:= ExponentsOfPcElement( pcgs, pcgs[ii]^pcgs[j], [ i .. lg ] );
          mat:= One( X );
          for jj in [ 1 .. lg-i+1 ] do
            mat:= mat * BaumClausenInfoDebug.makemat( rep1[jj], e )^exps[jj];
          od;
          if X * mat <>
             BaumClausenInfoDebug.makemat( rep2[ ii-i+1 ], e ) * X then
            return false;
          fi;
        od;
        return true;
    end ) );



#############################################################################
##
#M  BaumClausenInfo( <G> )  . . . . .  info about irreducible representations
##
#T generalize to characteristic p !!
##
InstallMethod( BaumClausenInfo,
    "for a (solvable) group",
    [ IsGroup ],
    function( G )
    local e,             # occurring roots of unity are `e'-th roots
          pcgs,          # Pcgs of `G'
          lg,            # length of `pcgs'
          cs,            # composition series of `G' corresp. to `pcgs'
          abel,          # position of abelian normal comp. subgroup
          ExtLinRep,     # local function
          indices,       # sizes of composition factors in `cs'
          linear,        # list of linear representations
          i,             # current position in the iteration: $G_i$
          p,             # size of current composition factor
          pexp,          # exponent vector of `pcgs[i]^p'
          root,          # value of an xtension
          roots,         # list of $p$-th roots (relative to `e')
          mulmoma,       # product of two monomial matrices
          poweval,       # representing matrix for power of generator
          pilinear,      # action of $g_1, \ldots, g_i$ on `linear'
          d, j, k, l,    # loop variables
          u, v, w,       # loop variables
          M,             #
          pos,           # position in a list
          nonlin,        # list of nonlinear representations
          pinonlin,      # action of $g_1, \ldots, g_i$ on `nonlin'
          Xlist,         # conjugating matrices:
                         # for $X = `Xlist[j][k]'$, we have
                         # $X \cdot {`nonlin[k]'}^{g_j} \cdot X^{-1} =
                         #    `nonlin[ pinonlin[j][k] ]'$
          min,           #
          minval,        #
          ssr,           #
          next,          #
          X,             # one matrix for `Xlist'
          nextlinear,    # extensions of `linear'
          nextnonlin1,   # nonlinear repr. arising from `linear'
          nextnonlin2,   # nonlinear repr. arising from `nonlin'
          pinextlinear,  # action of $g_1, \ldots, g_i$ on `nextlinear'
          pinextnonlin1, # action of $g_1, \ldots, g_i$ on `nextnonlin1'
          pinextnonlin2, # action of $g_1, \ldots, g_i$ on `nextnonlin2'
          nextXlist1,    # conjugating matrices for `nextnonlin1'
          nextXlist2,    # conjugating matrices for `nextnonlin2'
          cexp,          # exponent vector of `pcgs[i]^pcgs[j]'
          poli,          # list that encodes `pexp'
          rep,           # one representation
          D, C,          #
          value,         #
          image,         #
          used,          # boolean list
          Dpos1,         # positions of extension resp. induced repres.
                         # that arise from linear representations
          Dpos2,         # positions of extension resp. induced repres.
                         # that arise from nonlinear representations
          dim,           # dimension of the current representation
          invX,          # inverse of `X'
          D_gi,          #
          hom,           # homomorphism to adjust the composition series
          orb,           #
          Forb,          #
          sigma, pi,     # permutations needed in the fusion case
          constants,     # vector $(c_0, c_1, \ldots, c_{p-1})$
          kernel;        # kernel of `hom'

    if not IsSolvableGroup( G ) then
      Error( "<G> must be solvable" );
    fi;


    # Step 0:
    # Treat the case of the trivial group,
    # and initialize some variables.

    pcgs:= SpecialPcgs( G );
#T because I need a ``prime orders pcgs''
    lg:= Length( pcgs );

    if lg = 0 then
      return rec( pcgs     := pcgs,
                  kernel   := G,
                  exponent := 1,
                  nonlin   := [],
                  lin      := [ [] ]
                  );
    fi;

    cs:= PcSeries( pcgs );

    if HasExponent( G ) then
      e:= Exponent( G );
    else
      e:= Size(G);
#T better adjust on the fly
    fi;


    # Step 1:
    # If necessary then adjust the composition series of $G$
    # and get the largest factor group of $G$ that has an abelian normal
    # subgroup such that the factor group modulo this subgroup is
    # supersolvable.

    abel:= 1;
    while IsNormal( G, cs[ abel ] ) and not IsAbelian( cs[ abel ] ) do
      abel:= abel + 1;
    od;

    # If `cs[ abel ]' is abelian then we compute its representations first,
    # and then loop over the initial part of the composition series;
    # note that the factor group is supersolvable.
    # If `cs[ abel ]' is not abelian then we try to switch to a better
    # composition series, namely one through the derived subgroup of the
    # supersolvable residuum.

    if not IsNormal( G, cs[ abel ] ) then

      # We have reached a non-normal nonabelian composition subgroup
      # so we have to adjust the composition series.

      Info( InfoGroup, 2,
            "BaumClausenInfo: switching to a suitable comp. ser." );

      ssr:= SupersolvableResiduumDefault( G );
      hom:= NaturalHomomorphismByNormalSubgroupNC( G,
                DerivedSubgroup( ssr.ssr ) );

      # `SupersolvableResiduumDefault' contains a component `ds',
      # a list of subgroups such that any composition series through
      # `ds' from `G' down to the residuum is a chief series.
      pcgs:= [];
      for i in [ 2 .. Length( ssr.ds ) ] do
        j:= NaturalHomomorphismByNormalSubgroupNC( ssr.ds[ i-1 ], ssr.ds[i] );
        Append( pcgs, List( SpecialPcgs( ImagesSource( j ) ),
                            x -> PreImagesRepresentative( j, x ) ) );
      od;
      Append( pcgs, SpecialPcgs( ssr.ds[ Length( ssr.ds ) ]) );
      G:= ImagesSource( hom );
      pcgs:= List( pcgs, x -> ImagesRepresentative( hom, x ) );
      pcgs:= Filtered( pcgs, x -> Order( x ) <> 1 );
      pcgs:= PcgsByPcSequence( ElementsFamily( FamilyObj( G ) ), pcgs );
      cs:= PcSeries( pcgs );
      lg:= Length( pcgs );

      # The image of `ssr' under `hom' is abelian,
      # compute its position in the composition series.
      abel:= Position( cs, ImagesSet( hom, ssr.ssr ) );

      # If `G' is supersolvable then `abel = lg+1',
      # but the last *nontrivial* member of the chain is normal and abelian,
      # so we choose this group.
      # (Otherwise we would have the technical problem in step 4 that the
      # matrix `M' would be empty.)
      if lg < abel then
        abel:= lg;
      fi;

    fi;

    # Step 2:
    # Compute the representations of `cs[ abel ]',
    # each a list of images of $g_{abel}, \ldots, g_{lg}$.

    # The local function `ExtLinRep' computes the extensions of the
    # linear $G_{i+1}$-representations $F$ in the list `linear' to $G_i$.
    # The only condition that must be satisfied is that
    # $F(g_i)^p = F(g_i^p)$.
    # (Roughly speaking, we just compute $p$-th roots.)

    ExtLinRep:= function( i, linear, pexp, roots )

      local nextlinear, rep, j, shift;

      nextlinear:= [];
      if IsZero( pexp ) then

        # $g_i^p$ is the identity
        for rep in linear do
          for j in roots do
            Add( nextlinear, Concatenation( [ j ], rep ) );
          od;
        od;

      else

        pexp:= pexp{ [ i+1 .. lg ] };
#T cut this outside the function!
        for rep in linear do

          # Compute the value of `rep' on $g_i^p$.
          shift:= pexp * rep;

          if shift mod p <> 0 then
            # We must enlarge the exponent.
            Error("wrong exponent");
#T if not integral then enlarge the exponent!
#T (is this possible here at all?)
          fi;
          shift:= shift / p;
          for j in roots do
            Add( nextlinear, Concatenation( [ (j+shift) mod e ], rep ) );
          od;

        od;

      fi;

      return nextlinear;
    end;


    indices:= RelativeOrders( pcgs );
#T here set the exponent `e' to `indices[ lg ]' !
    Info( InfoGroup, 2,
          "BaumClausenInfo: There are ", lg, " steps" );

    linear:= List( [ 0 .. indices[lg]-1 ] * ( e / indices[lg] ),
                   x -> [ x ] );

    for i in [ lg-1, lg-2 .. abel ] do

      Info( InfoGroup, 2,
            "BaumClausenInfo: Compute repres. of step ", i,
            " (central case)" );

      p:= indices[i];

      # `pexp' describes $g_i^p$.
      pexp:= ExponentsOfRelativePower( pcgs,i);
# { ? } ??

      root:= e/p;
#T enlarge the exponent if necessary!
      roots:= [ 0, root .. (p-1)*root ];
      linear:= ExtLinRep( i, linear, pexp, roots );

    od;

    # We are done if $G$ is abelian.
    if abel = 1 then
      return rec( pcgs     := pcgs,
                  kernel   := TrivialSubgroup( G ),
                  exponent := e,
                  nonlin   := [],
                  lin      := linear
                  );
    fi;


    # Step 3:
    # Define some local functions.
    # (We did not need them for abelian groups.)

    # `mulmoma' returns the product of two monomial matrices.
    mulmoma:= function( a, b )
      local prod, i;
      prod:= rec( perm := b.perm{ a.perm },
                  diag := [] );
      for i in [ 1 .. Length( a.perm ) ] do
        prod.diag[ b.perm[i] ]:= ( b.diag[ b.perm[i] ] + a.diag[i] ) mod e;
      od;
      return prod;
    end;

    # `poweval' evaluates the representation `rep' on the $p$-th power of
    # the conjugating element.
    # This $p$-th power is described by `poli'.
    poweval:= function( rep, poli )
      local pow, i;
      if IsEmpty( poli ) then
        return rec( perm:= [ 1 .. Length( rep[1].perm ) ],
                    diag:= [ 1 .. Length( rep[1].perm ) ] * 0 );
      fi;
      pow:= rep[ poli[1] ];
      for i in [ 2 .. Length( poli ) ] do
        pow:= mulmoma( pow, rep[ poli[i] ] );
      od;
      return pow;
    end;


    # Step 4:
    # Compute the actions of $g_j$, $j < abel$, on the representations
    # of $G_{abel}$.
    # Let $g_i^{g_j} = \prod_{k=1}^n g_k^{\alpha_{ik}^j}$,
    # and set $A_j = [ \alpha_{ik}^j} ]_{i,k}$.
    # Then the representation that maps $g_i$ to the root $\zeta_e^{c_i}$
    # is mapped to the representation that has images exponents
    # $A_j * (c_1, \ldots, c_n)$ under $g_j$.

    Info( InfoGroup, 2,
          "BaumClausenInfo: Initialize actions on abelian normal subgroup" );

    pilinear:= [];
    for j in [ 1 .. abel-1 ] do

      # Compute the matrix $A_j$.
      M:= List( [ abel .. lg ],
                i -> ExponentsOfPcElement( pcgs, pcgs[i]^pcgs[j],
                                           [ abel .. lg ] ) );

      # Compute the permutation corresponding to the action of $g_j$.
      pilinear[j]:= List( linear,
                          rep -> Position( linear,
                                           List( M * rep, x -> x mod e ) ) );

    od;


    # Step 5:
    # Run up the composition series from `abel' to `1',
    # and compute extensions resp. induced representations.
    # For each index, we have to update `linear', `pilinear',
    # `nonlin', `pinonlin', and `Xlist'.

    nonlin   := [];
    pinonlin := [];
    Xlist    := [];

    for i in [ abel-1, abel-2 .. 1 ] do

      p:= indices[i];

      # `poli' describes $g_i^p$.
      #was pexp:= ExponentsOfPcElement( pcgs, pcgs[i]^p );
      pexp:= ExponentsOfRelativePower( pcgs, i );
      poli:= Concatenation( List( [ i+1 .. lg ],
                                  x -> List( [ 1 .. pexp[x] ],
                                             y -> x-i ) ) );

      # `p'-th roots of unity
      roots:= [ 0 .. p-1 ] * ( e/p );

      Info( InfoGroup, 2,
            "BaumClausenInfo: Compute repres. of step ", i );

      # Step A:
      # Compute representations of $G_i$ arising from *linear*
      # representations of $G_{i+1}$.

      used        := BlistList( [ 1 .. Length( linear ) ], [] );
      nextlinear  := [];
      nextnonlin1 := [];
      d           := 1;

      pexp:= pexp{ [ i+1 .. lg ] };

      # At position `d', store the position of either the first extension
      # of `linear[d]' in `nextlinear' or the position of the induced
      # representation of `linear[d]' in `nextnonlin1'.
      Dpos1:= [];

      while d <> fail do

        rep:= linear[d];
        used[d]:= true;

        # `root' is the value of `rep' on $g_i^p$.
        root:= ( pexp * rep ) mod e;

        if pilinear[i][d] = d then

          # `linear[d]' extends to $G_i$.
          Dpos1[d]:= Length( nextlinear ) + 1;

          # Take a `p'-th root.
          root:= root / p;
#T enlarge the exponent if necessary!

          for j in roots do
            Add( nextlinear, Concatenation( [ root+j ], rep ) );
          od;

        else

          # We must fuse the representations in the orbit of `d'
          # under `pilinear[i]';
          # so we construct the induced representation `D'.

          Dpos1[d]:= Length( nextnonlin1 ) + 1;

          D:= List( rep, x -> rec( perm := [ 1 .. p ],
                                   diag := [ x ]
                                  ) );
          pos:= d;
          for j in [ 2 .. p ] do

            pos:= pilinear[i][ pos ];
            for k in [ 1 .. Length( rep ) ] do
              D[k].diag[j]:= linear[ pos ][k];
            od;
            used[ pos ]:= true;
            Dpos1[ pos ]:= Length( nextnonlin1 ) + 1;

          od;

          Add( nextnonlin1,
               Concatenation( [ rec( perm := Concatenation( [p], [1..p-1]),
                                     diag := Concatenation( [ 1 .. p-1 ] * 0,
                                                            [ root ] ) ) ],
                              D ) );
          Assert( 2, BaumClausenInfoDebug.testrep( pcgs{ [ i .. lg ] },
                              nextnonlin1[ Length( nextnonlin1 ) ], e ),
                  Concatenation( "BaumClausenInfo: failed assertion in ",
                      "inducing linear representations ",
                      "(i = ", String( i ), ")\n" ) );

        fi;

        d:= Position( used, false, d );

      od;


      # Step B:
      # Now compute representations of $G_i$ arising from *nonlinear*
      # representations of $G_{i+1}$ (if there are some).

      used:= BlistList( [ 1 .. Length( nonlin ) ], [] );
      nextnonlin2:= [];
      if Length( nonlin ) = 0 then
        d:= fail;
      else
        d:= 1;
      fi;

      # At position `d', store the position of the first extension resp.
      # of the induced representation of `nonlin[d]'in `nextnonlin2'.
      Dpos2:= [];

      while d <> fail do

        used[d]:= true;
        rep:= nonlin[d];

        if pinonlin[i][d] = d then

          # The representation $F = `rep'$ has `p' different extensions.
          # For `X = Xlist[i][d]', we have $`rep ^ X' = `rep'^{g_i}$,
          # i.e., $X^{-1} F X = F^{g_i}$.
          # Representing matrix $F(g_i)$ is $c X$ with $c^p X^p = F(g_i^p)$,
          # so $c^p X^p.diag[k] = F(g_i^p).diag[k]$ for all $k$ ;
          # for determination of $c$ we look at `k = X^p.perm[1]'.

          X:= Xlist[i][d];
          image:= X.perm[1];
          value:= X.diag[ image ];
          for j in [ 2 .. p ] do

            image:= X.perm[ image ];
            value:= X.diag[ image ] + value;
            # now `image = X^j.perm[1]', `value = X^j.diag[ image ]'

          od;

          # Subtract this from $F(g_i^p).diag[k]$;
          # note that `image' is the image of 1 under `X^p', so also
          # under $F(g_i^p)$.
          value:= - value;
          image:= 1;
          for j in poli do
            image:= rep[j].perm[ image ];
            value:= rep[j].diag[ image ] + value;
          od;

          value:= ( value / p ) mod e;
#T enlarge the exponent if necessary!

          Dpos2[d]:= Length( nextnonlin2 ) + 1;

          # Compute the `p' extensions.
          for k in roots do
            Add( nextnonlin2, Concatenation(
                    [ rec( perm := X.perm,
                      diag := List( X.diag,
                             x -> ( x  + k + value ) mod e ) ) ], rep ) );
            Assert( 2, BaumClausenInfoDebug.testrep( pcgs{ [ i .. lg ] },
                                nextnonlin2[ Length( nextnonlin2 ) ], e ),
                    Concatenation( "BaumClausenInfo: failed assertion in ",
                        "extending nonlinear representations ",
                        "(i = ", String( i ), ")\n" ) );
          od;

        else

          # `$F$ = nonlin[d]' fuses with `p-1' partners given by the orbit
          # of `d' under `pinonlin[i]'.
          # The new irreducible representation of $G_i$ will be
          # $X Ind( F ) X^{-1}$ with $X$ the block diagonal matrix
          # consisting of blocks $X_{i,F}^{(k)}$ defined by
          # $X_{i,F}^{(0)} = Id$,
          # and $X_{i,F}^{(k)} = X_{i,\pi_i^{k-1} F} X_{i,F}^{(k-1)}$
          # for $k > 0$.

          # The matrix for $g_i$ in the induced representation $Ind( F )$ is
          # of the form
          #       | 0   F(g_i^p) |
          #       | I      0     |
          # Thus $X Ind(F) X^{-1} ( g_i )$ is the block diagonal matrix
          # consisting of the blocks
          # $X_{i,F}, X_{i,\pi_i F}, \ldots, X_{i,\pi_i^{p-2} F}$, and
          # $F(g_i^p) \cdot ( X_{i,F}^{(p-1)} )^{-1}$.

          dim:= Length( rep[1].diag );
          Dpos2[d]:= Length( nextnonlin2 ) + 1;

          # We make a copy of `rep' because we want to change it.
          D:= List( rep, record -> rec( perm := ShallowCopy( record.perm ),
                                        diag := ShallowCopy( record.diag )
                                       ) );

          # matrices for $g_j, i\< j \leq n$
          pos:= d;
          for j in [ 1 .. p-1 ] * dim do
            pos:= pinonlin[i][ pos ];
            for k in [ 1 .. Length( rep ) ] do
              Append( D[k].diag, nonlin[ pos ][k].diag );
              Append( D[k].perm, nonlin[ pos ][k].perm + j );
            od;

            used[ pos ]:= true;
            Dpos2[ pos ]:= Length( nextnonlin2 ) + 1;

          od;

          # The matrix of $g_i$ is a block-cycle with blocks
          # $X_{i,\pi_i^k(F)}$ for $0 \leq k \leq p-2$,
          # and $F(g_i^p) \cdot (X_{i,F}^{(p-1)})^{-1}$.

          X:= Xlist[i][d];      # $X_{i,F}$
          pos:= d;
          for j in [ 1 .. p-2 ] do
            pos:= pinonlin[i][ pos ];
            X:= mulmoma( Xlist[i][ pos ], X );
          od;

          # `invX' is the inverse of `X'.
          invX:= rec( perm := [], diag := [] );
          for j in [ 1 .. Length( X.diag ) ] do
            invX.perm[ X.perm[j] ]:= j;
            invX.diag[j]:= e - X.diag[ X.perm[j] ];
          od;
#T improve this using the {} operator!

          X:= mulmoma( poweval( rep, poli ), invX );
          D_gi:= rec( perm:= List( X.perm, x -> x  + ( p-1 ) * dim ),
                      diag:= [] );

          pos:= d;
          for j in [ 0 .. p-2 ] * dim do

            # $X_{i,\pi_i^j F}$
            Append( D_gi.diag, Xlist[i][ pos ].diag);
            Append( D_gi.perm, Xlist[i][ pos ].perm + j);
            pos:= pinonlin[i][ pos ];

          od;

          Append( D_gi.diag, X.diag );

          Add( nextnonlin2, Concatenation( [ D_gi ], D ) );
          Assert( 2, BaumClausenInfoDebug.testrep( pcgs{ [ i .. lg ] },
                              nextnonlin2[ Length( nextnonlin2 ) ], e ),
                  Concatenation( "BaumClausenInfo: failed assertion in ",
                      "inducing nonlinear representations ",
                      "(i = ", String( i ), ")\n" ) );

        fi;

        d:= Position( used, false, d );

      od;


      # Step C:
      # Compute `pilinear', `pinonlin', and `Xlist'.

      pinextlinear  := [];
      pinextnonlin1 := [];
      nextXlist1    := [];

      pinextnonlin2 := [];
      nextXlist2    := [];

      for j in [ 1 .. i-1 ] do

        pinextlinear[j]  := [];
        pinextnonlin1[j] := [];
        nextXlist1[j]    := [];

        # `cexp' describes $g_i^{g_j}$.
        cexp:= ExponentsOfPcElement( pcgs, pcgs[i]^pcgs[j], [ i .. lg ] );

        # Compute `pilinear', and the parts of `pinonlin', `Xlist'
        # arising from *linear* representations for the next step,
        # that is, compute the action of $g_j$ on `nextlinear' and
        # `nextnonlin1'.

        for k in [ 1 .. Length( linear ) ] do

          if pilinear[i][k] = k then

            # Let $F = `linear[k]'$ extend to
            # $D = D_0, D_1, \ldots, D_{p-1}$,
            # $C$ the first extension of $\pi_j(F)$.
            # We have $D( g_i^{g_j} ) = D^{g_j}(g_i) = ( C \chi^l )(g_i)$
            # where $\chi^l(g_i)$ is the $l$-th power of the chosen
            # primitive $p$-th root of unity.

            D:= nextlinear[ Dpos1[k] ];

            # `pos' is the position of $C$ in `nextlinear'.
            pos:= Dpos1[ pilinear[j][k] ];
            l:= ( (  cexp * D                   # $D( g_i^{g_j} )$
                     - nextlinear[ pos ][1] )   # $C(g_i)$
                  * p / e ) mod p;

            for u in [ 0 .. p-1 ] do
              Add( pinextlinear[j], pos + ( ( l + u * cexp[1] ) mod p ) );
            od;

          elif not IsBound( pinextnonlin1[j][ Dpos1[k] ] ) then

            # $F$ fuses with its conjugates under $g_i$,
            # the conjugating matrix describing the action of $g_j$
            # is a permutation matrix.
            # Let $D = F^{g_j}$, then the permutation corresponds to
            # the mapping between the lists
            # $[ D, (F^{g_i})^{g_j}, \ldots, (F^{g_i^{p-1}})^{g_j} ]$
            # and $[ D, D^{g_i}, \ldots, D^{g_i^{p-1}} ]$;
            # The constituents in the first list are the images of
            # the induced representation of $F$ under $g_j$,
            # and those in the second list are the constituents of the
            # induced representation of $D$.

            # While `u' runs from $1$ to $p$,
            # `pos' runs over the positions of $(F^{g_i^u})^{g_j}$ in
            # `linear'.
            # `orb' is the list of positions of the $(F^{g_j})^{g_i^u}$,
            # cyclically permuted such that the smallest entry is the
            # first.

            pinextnonlin1[j][ Dpos1[k] ]:= Dpos1[ pilinear[j][k] ];
            pos:= pilinear[j][k];
            orb:= [ pos ];
            min:= 1;
            minval:= pos;
            for u in [ 2 .. p ] do
              pos:= pilinear[i][ pos ];
              orb[u]:= pos;
              if pos < minval then
                minval:= pos;
                min:= u;
              fi;
            od;
            if 1 < min then
              orb:= Concatenation( orb{ [ min .. p ] },
                                   orb{ [ 1 .. min-1 ] } );
            fi;

            # Compute the conjugating matrix `X'.
            # Let $C$ be the stored representation $\tau_j D$
            # equivalent to $D^{g_j}$.
            # Compute the position of $C$ in `pinextnonlin1'.

            C:= nextnonlin1[ pinextnonlin1[j][ Dpos1[k] ] ];
            D:= nextnonlin1[ Dpos1[k] ];

            # `sigma' is the bijection of constituents in the restrictions
            # of $D$ and $\tau_j D$ to $G_{i-1}$.
            # More precisely, $\pi_j(\pi_i^{u-1} F) = \Phi_{\sigma(u-1)}$.
            sigma:= [];
            pos:= k;
            for u in [ 1 .. p ] do
              sigma[u]:= Position( orb, pilinear[j][ pos ] );
              pos:= pilinear[i][ pos ];
            od;

            # Compute $\pi = \sigma^{-1} (1,2,\ldots,p) \sigma$.
            pi:= [];
            pi[ sigma[p] ]:= sigma[1];
            for u in [ 1 .. p-1 ] do
              pi[ sigma[u] ]:= sigma[ u+1 ];
            od;

            # Compute the values $c_{\pi^u(0)}$, for $0 \leq u \leq p-1$.
            # Note that $c_0 = 1$.
            # (Here we encode of course the exponents.)
            constants:= [ 0 ];
            l:= 1;

            for u in [ 1 .. p-1 ] do

              # Compute $c_{\pi^u(0)}$.
              # (We have $`l' = 1 + \pi^{u-1}(0)$.)
              # Note that $B_u = [ [ 1 ] ]$ for $0\leq u\leq p-2$,
              # and $B_{p-1} = \Phi_0(g_i^p)$.

              # Next we compute the image under $A_{\pi^{u-1}(0)}$;
              # this matrix is in the $(\pi^{u-1}(0)+1)$-th column block
              # and in the $(\pi^u(0)+1)$-th row block of $D^{g_j}$.
              # Since we do not have this matrix explicitly,
              # we use the conjugate representation and the action
              # encoded by `cexp'.
              # Note the necessary initial shift because we use the
              # whole representation $D$ and not a single constituent;
              # so we shift by $\pi^u(0)+1$.
#T `perm' is nontrivial only for v = 1, this should make life easier.
              value:= 0;
              image:= pi[l];
              for v in [ 1 .. lg-i+1 ] do
                for w in [ 1 .. cexp[v] ] do
                  image:= D[v].perm[ image ];
                  value:= value + D[v].diag[ image ];
                od;
              od;

              # Next we divide by the corresponding value in
              # the image of the first standard basis vector under
              # $B_{\sigma\pi^{u-1}(0)}$.
              value:= value - C[1].diag[ sigma[l] ];
              constants[ pi[l] ]:= ( constants[l] - value ) mod e;
              l:= pi[l];

            od;

            # Put the conjugating matrix together.
            X:= rec( perm := [],
                     diag := constants );
            for u in [ 1 .. p ] do
              X.perm[ sigma[u] ]:= u;
            od;

            Assert( 2, BaumClausenInfoDebug.checkconj( pcgs, i, lg, j,
                         nextnonlin1[ Dpos1[k] ],
                         nextnonlin1[ pinextnonlin1[j][ Dpos1[k] ] ],
                         X, e ),
                  Concatenation( "BaumClausenInfo: failed assertion on ",
                      "conjugating matrices for linear repres. ",
                      "(i = ", String( i ), ")\n" ) );
            nextXlist1[j][ Dpos1[k] ]:= X;

          fi;

        od;


        # Compute the remaining parts of `pinonlin' and `Xlist' for
        # the next step, namely for those *nonlinear* representations
        # arising from *nonlinear* ones.

        nextXlist2[j]    := [];
        pinextnonlin2[j] := [];

        # `cexp' describes $g_i^{g_j}$.
        cexp:= ExponentsOfPcElement( pcgs, pcgs[i]^pcgs[j], [ i .. lg ] );

        # Compute the action of $g_j$ on `nextnonlin2'.

        for k in [ 1 .. Length( nonlin ) ] do

          if pinonlin[i][k] = k then

            # Let $F = `nonlin[k]'$ extend to
            # $D = D_0, D_1, \ldots, D_{p-1}$,
            # $C$ the first extension of $\pi_j(F)$.
            # We have $X_{j,F} \cdot F^{g_j} = \pi_j(F) \cdot X_{j,F}$,
            # thus $X_{j,F} \cdot D( g_i^{g_j} )
            # = X_{j,F} \cdot D^{g_j}(g_i)
            # = ( C \chi^l )(g_i) \cdot X_{j,F}$
            # where $\chi^l(g_i)$ is the $l$-th power of the chosen
            # primitive $p$-th root of unity.

            D:= nextnonlin2[ Dpos2[k] ];

            # `pos' is the position of $C$ in `nextnonlin2'.
            pos:= Dpos2[ pinonlin[j][k] ];

            # Find a nonzero entry in $X_{j,F} \cdot D( g_i^{g_j} )$.
            image:= Xlist[j][k].perm[1];
            value:= Xlist[j][k].diag[ image ];
            for u in [ 1 .. lg-i+1 ] do
              for v in [ 1 .. cexp[u] ] do
                image:= D[u].perm[ image ];
                value:= value + D[u].diag[ image ];
              od;
            od;

            # Subtract the corresponding value in $C(g_i) \cdot X_{j,F}$.
            C:= nextnonlin2[ pos ];
            Assert( 2, image = Xlist[j][k].perm[ C[1].perm[1] ],
                    "BaumClausenInfo: failed assertion on conj. matrices" );
            value:= value -
                ( C[1].diag[ C[1].perm[1] ] + Xlist[j][k].diag[ image ] );
            l:= ( value * p / e ) mod p;

            for u in [ 0 .. p-1 ] do
              pinextnonlin2[j][ Dpos2[k] + u ]:=
                     pos + ( ( l + u * cexp[1] ) mod p );
              nextXlist2[j][ Dpos2[k] + u ]:= Xlist[j][k];
            od;

            Assert( 2, BaumClausenInfoDebug.checkconj( pcgs, i, lg, j,
                         nextnonlin2[ Dpos2[k] ],
                         nextnonlin2[ pinextnonlin2[j][ Dpos2[k] ] ],
                         Xlist[j][k], e ),
                  Concatenation( "BaumClausenInfo: failed assertion on ",
                      "conjugating matrices for nonlinear repres. ",
                      "(i = ", String( i ), ")\n" ) );

          elif not IsBound( pinextnonlin2[j][ Dpos2[k] ] ) then

            # $F$ fuses with its conjugates under $g_i$, yielding $D$.

            dim:= Length( nonlin[k][1].diag );

            # Let $C$ be the stored representation $\tau_j D$
            # equivalent to $D^{g_j}$.
            # Compute the position of $C$ in `pinextnonlin2'.
            pinextnonlin2[j][ Dpos2[k] ]:= Dpos2[ pinonlin[j][k] ];

            C:= nextnonlin2[ pinextnonlin2[j][ Dpos2[k] ] ];
            D:= nextnonlin2[ Dpos2[k] ];

            # Compute the positions of the constituents;
            # `orb[k]' is the position of $\Phi_{k-1}$ in `nonlin'.
            pos:= pinonlin[j][k];
            orb:= [ pos ];
            min:= 1;
            minval:= pos;
            for u in [ 2 .. p ] do
              pos:= pinonlin[i][ pos ];
              orb[u]:= pos;
              if pos < minval then
                minval:= pos;
                min:= u;
              fi;
            od;
            if 1 < min then
              orb:= Concatenation( orb{ [ min .. p ] },
                                   orb{ [ 1 .. min-1 ] } );
            fi;

            # `sigma' is the bijection of constituents in the restrictions
            # of $D$ and $\tau_j D$ to $G_{i-1}$.
            # More precisely, $\pi_j(\pi_i^{u-1} F) = \Phi_{\sigma(u-1)}$.
            sigma:= [];
            pos:= k;
            for u in [ 1 .. p ] do
              sigma[u]:= Position( orb, pinonlin[j][ pos ] );
              pos:= pinonlin[i][ pos ];
            od;

            # Compute $\pi = \sigma^{-1} (1,2,\ldots,p) \sigma$.
            pi:= [];
            pi[ sigma[p] ]:= sigma[1];
            for u in [ 1 .. p-1 ] do
              pi[ sigma[u] ]:= sigma[ u+1 ];
            od;

            # Compute the positions of the constituents
            # $F_0, F_{\pi(0)}, \ldots, F_{\pi^{p-1}(0)}$.
            Forb:= [ k ];
            pos:= k;
            for u in [ 2 .. p ] do
              pos:= pinonlin[i][ pos ];
              Forb[u]:= pos;
            od;

            # Compute the values $c_{\pi^u(0)}$, for $0 \leq u \leq p-1$.
            # Note that $c_0 = 1$.
            # (Here we encode of course the exponents.)
            constants:= [ 0 ];
            l:= 1;

            for u in [ 1 .. p-1 ] do

              # Compute $c_{\pi^u(0)}$.
              # (We have $`l' = 1 + \pi^{u-1}(0)$.)
              # Note that $B_u = X_{j,\pi_j^u \Phi_0}$ for $0\leq u\leq p-2$,
              # and $B_{p-1} =
              #      \Phi_0(g_i^p) \cdot ( X_{j,\Phi_0}^{(p-1)} )^{-1}$

              # First we get the image and diagonal value of
              # the first standard basis vector under $X_{j,\pi^u(0)}$.
              image:= Xlist[j][ Forb[ pi[l] ] ].perm[1];
              value:= Xlist[j][ Forb[ pi[l] ] ].diag[ image ];

              # Next we compute the image under $A_{\pi^{u-1}(0)}$;
              # this matrix is in the $(\pi^{u-1}(0)+1)$-th column block
              # and in the $(\pi^u(0)+1)$-th row block of $D^{g_j}$.
              # Since we do not have this matrix explicitly,
              # we use the conjugate representation and the action
              # encoded by `cexp'.
              # Note the necessary initial shift because we use the
              # whole representation $D$ and not a single constituent;
              # so we shift by `dim' times $\pi^u(0)+1$.
              image:= dim * ( pi[l] - 1 ) + image;
              for v in [ 1 .. lg-i+1 ] do
                for w in [ 1 .. cexp[v] ] do
                  image:= D[v].perm[ image ];
                  value:= value + D[v].diag[ image ];
                od;
              od;

              # Next we divide by the corresponding value in
              # the image of the first standard basis vector under
              # $B_{\sigma\pi^{u-1}(0)} X_{j,\pi^{u-1}(0)}$.
              # Note that $B_v$ is in the $(v+2)$-th row block for
              # $0 \leq v \leq p-2$, in the first row block for $v = p-1$,
              # and in the $(v+1)$-th column block of $C$.
              v:= sigma[l];
              if v = p then
                image:= C[1].perm[1];
              else
                image:= C[1].perm[ v*dim + 1 ];
              fi;
              value:= value - C[1].diag[ image ];
              image:= Xlist[j][ Forb[l] ].perm[ image - ( v - 1 ) * dim ];
              value:= value - Xlist[j][ Forb[l] ].diag[ image ];
              constants[ pi[l] ]:= ( constants[l] - value ) mod e;
              l:= pi[l];

            od;

            # Put the conjugating matrix together.
            X:= rec( perm:= [],
                     diag:= [] );
            pos:= k;
            for u in [ 1 .. p ] do
              Append( X.diag, List( Xlist[j][ pos ].diag,
                                    x -> ( x + constants[u] ) mod e ) );
              X.perm{ [ ( sigma[u] - 1 )*dim+1 .. sigma[u]*dim ] }:=
                  Xlist[j][ pos ].perm + (u-1) * dim;
              pos:= pinonlin[i][ pos ];
            od;

            Assert( 2, BaumClausenInfoDebug.checkconj( pcgs, i, lg, j,
                         nextnonlin2[ Dpos2[k] ],
                         nextnonlin2[ pinextnonlin2[j][ Dpos2[k] ] ],
                         X, e ),
                  Concatenation( "BaumClausenInfo: failed assertion on ",
                      "conjugating matrices for nonlinear repres. ",
                      "(i = ", String( i ), ")\n" ) );
            nextXlist2[j][ Dpos2[k] ]:= X;

          fi;

        od;

      od;

      # Finish the update for the next index.
      linear   := nextlinear;
      pilinear := pinextlinear;

      nonlin   := Concatenation( nextnonlin1, nextnonlin2 );
      pinonlin := List( [ 1 .. i-1 ],
                       j -> Concatenation( pinextnonlin1[j],
                         pinextnonlin2[j] + Length( pinextnonlin1[j] ) ) );
      Xlist    := List( [ 1 .. i-1 ],
                    j -> Concatenation( nextXlist1[j], nextXlist2[j] ) );

    od;


    # Step 6: If necessary transfer the representations back to the
    #         original group.

    if     IsBound( hom )
       and not IsTrivial( KernelOfMultiplicativeGeneralMapping( hom ) ) then
      Info( InfoGroup, 2,
            "BaumClausenInfo: taking preimages in the original group" );

      kernel:= KernelOfMultiplicativeGeneralMapping( hom );
      k:= Pcgs( kernel );
      pcgs:= PcgsByPcSequence( ElementsFamily( FamilyObj( kernel ) ),
               Concatenation( List( pcgs,
                                    x -> PreImagesRepresentative( hom, x ) ),
                              k ) );
      k:= ListWithIdenticalEntries( Length( k ), 0 );

      linear:= List( linear, rep -> Concatenation( rep, k ) );

      for rep in nonlin do
        dim:= Length( rep[1].perm );
        M:= rec( perm:= [ 1 .. dim ],
                 diag:= [ 1 .. dim ] * 0 );
        for i in k do
          Add( rep, M );
        od;
      od;

    else
      kernel:= TrivialSubgroup( G );
    fi;

    # Return the result (for nonabelian groups).
    return Immutable( rec( pcgs     := pcgs,
                           kernel   := kernel,
                           exponent := e,
                           nonlin   := nonlin,
                           lin      := linear
                          ) );
    end );


#############################################################################
##
#F  IrreducibleRepresentationsByBaumClausen( <G> )  .  for a supersolv. group
##
BindGlobal( "IrreducibleRepresentationsByBaumClausen", function( G )
    local mrep,    # list of images lists for the result
          info,    # result of `BaumClausenInfo'
          lg,      # composition length of `G'
          rep,     # loop over the representations
          gcd,     # g.c.d. of the exponents in `rep'
          Ee,      # complex root of unity needed for `rep'
          images,  # one list of images
          dim,     # current dimension
          i, k,    # loop variabes
          mat;     # one representing matrix

    mrep:= [];
    info:= BaumClausenInfo( G );
    lg:= Length( info.pcgs );
    
    if info.lin=[[]] then # trivial group
        return [GroupHomomorphismByImagesNC(G,Group([[1]]),[],[])];
    fi;
    
    # Compute the images of linear representations on the pcgs.
    for rep in info.lin do
      gcd := Gcd( rep );
      if gcd = 0 then
        Add( mrep, List( rep, x -> [ [ 1 ] ] ) );
      else
        gcd:= GcdInt( gcd, info.exponent );
        Ee:= E( info.exponent / gcd );
        Add( mrep, List( rep / gcd, x -> [ [ Ee^x ] ] ) );
      fi;
    od;

    # Compute the images of nonlinear representations on the pcgs.
    for rep in info.nonlin do
      images:= [];
      dim:= Length( rep[1].perm );
      gcd:= GcdInt( Gcd( List( rep, x -> Gcd( x.diag ) ) ), info.exponent );
      Ee:= E( info.exponent / gcd );
      for i in [ 1 .. lg ] do
        mat:= NullMat( dim, dim, Rationals );
        for k in [ 1 .. dim ] do
          mat[k][ rep[i].perm[k] ]:=
              Ee^( rep[i].diag[ rep[i].perm[k] ] / gcd );
        od;
        images[i]:= mat;
      od;
      Add( mrep, images );
    od;

    return List( mrep, images -> GroupHomomorphismByImagesNC( G,
                     GroupByGenerators( images ), info.pcgs, images ) );
    end );


#############################################################################
##
#M  IrreducibleRepresentations( <G> ) . for an abelian by supersolvable group
##
InstallMethod( IrreducibleRepresentations,
    "(abelian by supersolvable) finite group",
    [ IsGroup and IsFinite ], 1, # higher than Dixon's method
    function( G )
    if IsAbelian( SupersolvableResiduum( G ) ) then
      return IrreducibleRepresentationsByBaumClausen( G );
    else
      TryNextMethod();
    fi;
    end );


#############################################################################
##
#M  IrreducibleRepresentations( <G>, <F> )  . . for a group and `Cyclotomics'
##
InstallMethod( IrreducibleRepresentations,
    "finite group, Cyclotomics",
    [ IsGroup and IsFinite, IsCyclotomicCollection and IsField ],
    function( G, F )
    if F <> Cyclotomics then
      TryNextMethod();
    else
      return IrreducibleRepresentations( G );
    fi;
    end );


#############################################################################
##
#M  IrreducibleRepresentations( <G>, <f> )
##
InstallMethod( IrreducibleRepresentations,
    "for a finite group over a finite field",
    [ IsGroup and IsFinite, IsField and IsFinite ],
    function( G, f )
    local md, hs, gens, M, mats, H, hom;

    md := IrreducibleModules( G, f, 0 );
    gens:=md[1];
    md:=md[2];
    hs := [];
    for M in md do
        mats := M.generators;
        H    := Group( mats, IdentityMat( M.dimension, f ) );
        hom  := GroupHomomorphismByImagesNC( G, H, gens, mats );
        Add( hs, hom );
    od;
    return hs;
    end );


#############################################################################
##
#M  IrrBaumClausen( <G>)  . . . .  irred. characters of a supersolvable group
##
InstallMethod( IrrBaumClausen,
    "for a (solvable) group",
    [ IsGroup ],
    function( G )
    local mulmoma,        # local function  to multiply monomial matrices
          ccl,            # conjugacy classes of `G'
          tbl,            # character table of `G'
          info,           # result of `BaumClausenInfo'
          pcgs,           # value of `info.pcgs'
          lg,             # composition length
          evl,            # list encoding exponents of class representatives
          i, j, k,        # loop variables
          exps,           # exponent vector of a group element
          t,              # intermediate representation value
          irreducibles,   # list of irreducible characters
          rep,            # loop over the representations
          gcd,            # g.c.d. of the exponents in `rep'
          q,              # 
          Ee,             # complex root of unity needed for `rep'
          chi,            # one character values list
          deg,            # character degree
          idmat,          # identity matrix
          trace;          # trace of a matrix

    mulmoma:= function( a, b )
      local prod, i;
      prod:= rec( perm := b.perm{ a.perm },
                  diag := [] );
      for i in [ 1 .. deg ] do
        prod.diag[ b.perm[i] ]:= b.diag[ b.perm[i] ] + a.diag[i];
      od;
      return prod;
    end;

    tbl:= CharacterTable( G );
    ccl:= ConjugacyClasses( tbl );
    SetExponent( G, Exponent( tbl ) );
    info:= BaumClausenInfo( G );

    # The trivial group does not admit matrix arithmetic for evaluations.
    if IsTrivial( G ) then
      return [ Character( G, [ 1 ] ) ];
    fi;

    pcgs:= info.pcgs;
    lg:= Length( pcgs );

    exps:= List( ccl,
                 c -> ExponentsOfPcElement( pcgs, Representative( c ) ) );

    # Compute the linear irreducibles.
    # Compute the roots of unity only once for all linear characters.
    # ($q$-th roots suffice, where $q$ divides the number of linear
    # characters and the known exponent; we do *not* compute the smallest
    # possible roots for each representation.)
    q:= Gcd( info.exponent, Length( info.lin ) );
    gcd:= info.exponent / q;
    Ee:= E(q);
    Ee:= List( [ 0 .. q-1 ], i -> Ee^i );
    irreducibles:= List( info.lin, rep ->
        Character( tbl, Ee{ ( ( exps * rep ) / gcd mod q ) + 1 } ) );

    # Compute the nonlinear irreducibles.
    if not IsEmpty( info.nonlin ) then
      evl:= [];
      for i in [ 2 .. Length( ccl ) ] do
        t:= [];
        for j in [ 1 .. lg ] do
          for k in [ 1 .. exps[i][j] ] do
            Add( t, j );
          od;
        od;
        evl[ i-1 ]:= t;
      od;
      for rep in info.nonlin do
        gcd:= GcdInt( Gcd( List( rep, x -> Gcd( x.diag ) ) ), info.exponent );
        Ee:= E( info.exponent / gcd );
        deg:= Length( rep[1].perm );
        chi:= [ deg ];
        idmat:= rec( perm := [ 1 .. deg ], diag := [ 1 .. deg ] * 0 );
        for j in evl do

          # Compute the value of the representation at the representative.
          t:= idmat;
          for k in j do
            t:= mulmoma( t, rep[k] );
          od;

          # Compute the character value.
          trace:= 0;
          for k in [ 1 .. deg ] do
            if t.perm[k] = k then
              trace:= trace + Ee^( t.diag[k] / gcd );
            fi;
          od;
          Add( chi, trace );

        od;
        Add( irreducibles, Character( tbl, chi ) );
      od;
    fi;

    # Return the result.
    return irreducibles;
    end );


#############################################################################
##
#F  InducedRepresentationImagesRepresentative( <rep>, <H>, <R>, <g> )
##
##  Let $<rep>_H$ denote the restriction of the group homomorphism <rep> to
##  the group <H>, and $\phi$ the induced representation of $<rep>_H$ to $G$,
##  where <R> is a transversal of <H> in $G$.
##  `InducedRepresentationImagesRepresentative' returns the image of the
##  element <g> of $G$ under $\phi$.
##
InstallGlobalFunction( InducedRepresentationImagesRepresentative,
    function( rep, H, R, g )
    local len, blocks, i, k, kinv, j;

    len:= Length( R );
    blocks:= [];

    for i in [ 1 .. len ] do
      k:= R[i] * g;
      kinv:= Inverse( k );
      j:= PositionProperty( R, r -> r * kinv in H );
      blocks[i]:= [ i, j, ImagesRepresentative( rep, k / R[j] ) ];
    od;

    return BlockMatrix( blocks, len, len );
end );


#############################################################################
##
#F  InducedRepresentation( <rep>, <G> ) . . . . induced matrix representation
#F  InducedRepresentation( <rep>, <G>, <R> )
#F  InducedRepresentation( <rep>, <G>, <R>, <H> )
##
##  Let <rep> be a matrix representation of the group $H$, which is a
##  subgroup of the group <G>.
##  `InducedRepresentation' returns the induced matrix representation of <G>.
##
##  The optional third argument <R> is a right transversal of $H$ in <G>.
##  If the fourth optional argument <H> is given then it must be a subgroup
##  of the source of <rep>, and the induced representation of the restriction
##  of <rep> to <H> is computed.
##
InstallGlobalFunction( InducedRepresentation, function( arg )
    local rep, G, H, R, gens, images, map;

    # Get and check the arguments.
    if   Length( arg ) = 2 and IsGroupHomomorphism( arg[1] )
                           and IsGroup( arg[2] ) then
      rep := arg[1];
      G   := arg[2];
      H   := Source( rep );
      R   := RightTransversal( G, H );

    elif Length( arg ) = 3 and IsGroupHomomorphism( arg[1] )
                           and IsGroup( arg[2] )
                           and IsHomogeneousList( arg[3] ) then
      rep := arg[1];
      G   := arg[2];
      R   := arg[3];
      H   := Source( rep );

    elif Length( arg ) = 4 and IsGroupHomomorphism( arg[1] )
                           and IsGroup( arg[2] )
                           and IsHomogeneousList( arg[3] )
                           and IsGroup( arg[4] ) then
      rep := arg[1];
      G   := arg[2];
      R   := arg[3];
      H   := arg[4];

    else
      Error( "usage: InducedRepresentation(<rep>,<G>[,<R>[,<H>]])" );
    fi;

    # Handle a trivial case.
    if Length( R ) = 1 then
      return rep;
    fi;

    # Construct the images of the generators of <G>.
    gens:= GeneratorsOfGroup( G );
    images:= List( gens,
        g -> InducedRepresentationImagesRepresentative( rep, H, R, g ) );

    # Construct and return the homomorphism.
    map:= GroupHomomorphismByImagesNC( G, GroupByGenerators( images ),
                                     gens, images );
    SetIsSurjective( map, true );
    return map;
end );


#############################################################################
##
#M  <rep> ^ <G>
##
InstallOtherMethod( \^,
    "for group homomorphism and group (induction)",
    [ IsGroupHomomorphism, IsGroup ],
    function( rep, G )
    if IsMatrixGroup( Range( rep ) ) and IsSubset( Source( rep ), G ) then
      return InducedRepresentation( rep, G );
    else
      TryNextMethod();
    fi;
    end );


#############################################################################
##
#E