  
  [1X7 [33X[0;0YHigher level methods for pcp-groups[133X[101X
  
  [33X[0;0YThis  is  a  description  of  some  higher level functions of the [5XPolycyclic[105X
  package  of  GAP 4. Throughout this chapter we let [3XG[103X be a pc-presented group
  and  we consider algorithms for subgroups [3XU[103X and [3XV[103X of [3XG[103X. For background and a
  description of the underlying algorithms we refer to [Eic01a].[133X
  
  
  [1X7.1 [33X[0;0YSubgroup series in pcp-groups[133X[101X
  
  [33X[0;0YMany  algorithm  for  pcp-groups work by induction using some series through
  the  group.  In  this  section  we  provide  a  number  of useful series for
  pcp-groups. An [13Xefa series[113X is a normal series with elementary or free abelian
  factors.  See  [Eic00]  for  outlines  on  the algorithms of a number of the
  available series.[133X
  
  [1X7.1-1 PcpSeries[101X
  
  [33X[1;0Y[29X[2XPcpSeries[102X( [3XU[103X ) [32X function[133X
  
  [33X[0;0Yreturns the polycyclic series of [3XU[103X defined by an igs of [3XU[103X.[133X
  
  [1X7.1-2 EfaSeries[101X
  
  [33X[1;0Y[29X[2XEfaSeries[102X( [3XU[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns a normal series of [3XU[103X with elementary or free abelian factors.[133X
  
  [1X7.1-3 SemiSimpleEfaSeries[101X
  
  [33X[1;0Y[29X[2XSemiSimpleEfaSeries[102X( [3XU[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  an  efa  series  of  [3XU[103X  such  that  every  factor  in the series is
  semisimple as a module for [3XU[103X over a finite field or over the rationals.[133X
  
  [1X7.1-4 DerivedSeriesOfGroup[101X
  
  [33X[1;0Y[29X[2XDerivedSeriesOfGroup[102X( [3XU[103X ) [32X method[133X
  
  [33X[0;0Ythe derived series of [3XU[103X.[133X
  
  [1X7.1-5 RefinedDerivedSeries[101X
  
  [33X[1;0Y[29X[2XRefinedDerivedSeries[102X( [3XU[103X ) [32X function[133X
  
  [33X[0;0Ythe  derived  series of [3XU[103X refined to an efa series such that in each abelian
  factor of the derived series the free abelian factor is at the top.[133X
  
  [1X7.1-6 RefinedDerivedSeriesDown[101X
  
  [33X[1;0Y[29X[2XRefinedDerivedSeriesDown[102X( [3XU[103X ) [32X function[133X
  
  [33X[0;0Ythe  derived  series of [3XU[103X refined to an efa series such that in each abelian
  factor of the derived series the free abelian factor is at the bottom.[133X
  
  [1X7.1-7 LowerCentralSeriesOfGroup[101X
  
  [33X[1;0Y[29X[2XLowerCentralSeriesOfGroup[102X( [3XU[103X ) [32X method[133X
  
  [33X[0;0Ythe  lower  central  series  of  [3XU[103X.  If  [3XU[103X does not have a largest nilpotent
  quotient group, then this function may not terminate.[133X
  
  [1X7.1-8 UpperCentralSeriesOfGroup[101X
  
  [33X[1;0Y[29X[2XUpperCentralSeriesOfGroup[102X( [3XU[103X ) [32X method[133X
  
  [33X[0;0Ythe  upper  central series of [3XU[103X. This function always terminates, but it may
  terminate at a proper subgroup of [3XU[103X.[133X
  
  [1X7.1-9 TorsionByPolyEFSeries[101X
  
  [33X[1;0Y[29X[2XTorsionByPolyEFSeries[102X( [3XU[103X ) [32X function[133X
  
  [33X[0;0Yreturns an efa series of [3XU[103X such that all torsion-free factors are at the top
  and  all finite factors are at the bottom. Such a series might not exist for
  [3XU[103X and in this case the function returns fail.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfSomePcpGroups(5);[127X[104X
    [4X[28XPcp-group with orders [ 2, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XIgs(G);[127X[104X
    [4X[28X[ g1, g2, g3, g4 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XPcpSeries(G);[127X[104X
    [4X[28X[ Pcp-group with orders [ 2, 0, 0, 0 ],[128X[104X
    [4X[28X  Pcp-group with orders [ 0, 0, 0 ],[128X[104X
    [4X[28X  Pcp-group with orders [ 0, 0 ],[128X[104X
    [4X[28X  Pcp-group with orders [ 0 ],[128X[104X
    [4X[28X  Pcp-group with orders [  ] ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XList( PcpSeries(G), Igs );[127X[104X
    [4X[28X[ [ g1, g2, g3, g4 ], [ g2, g3, g4 ], [ g3, g4 ], [ g4 ], [  ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlgorithms  for  pcp-groups  often use an efa series of [22XG[122X and work down over
  the  factors  of  this series. Usually, pcp's of the factors are more useful
  than the actual factors. Hence we provide the following.[133X
  
  [1X7.1-10 PcpsBySeries[101X
  
  [33X[1;0Y[29X[2XPcpsBySeries[102X( [3Xser[103X[, [3Xflag[103X] ) [32X function[133X
  
  [33X[0;0Yreturns  a  list of pcp's corresponding to the factors of the series. If the
  parameter  [3Xflag[103X  is  present  and  equals  the  string  [21Xsnf[121X,  then  each pcp
  corresponds to a decomposition of the abelian groups into direct factors.[133X
  
  [1X7.1-11 PcpsOfEfaSeries[101X
  
  [33X[1;0Y[29X[2XPcpsOfEfaSeries[102X( [3XU[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns a list of pcps corresponding to an efa series of [3XU[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfSomePcpGroups(5);[127X[104X
    [4X[28XPcp-group with orders [ 2, 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XPcpsBySeries( DerivedSeriesOfGroup(G));[127X[104X
    [4X[28X[ Pcp [ g1, g2, g3, g4 ] with orders [ 2, 2, 2, 2 ],[128X[104X
    [4X[28X  Pcp [ g2^-2, g3^-2, g4^2 ] with orders [ 0, 0, 4 ],[128X[104X
    [4X[28X  Pcp [ g4^8 ] with orders [ 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XPcpsBySeries( RefinedDerivedSeries(G));[127X[104X
    [4X[28X[ Pcp [ g1, g2, g3 ] with orders [ 2, 2, 2 ],[128X[104X
    [4X[28X  Pcp [ g4 ] with orders [ 2 ],[128X[104X
    [4X[28X  Pcp [ g2^2, g3^2 ] with orders [ 0, 0 ],[128X[104X
    [4X[28X  Pcp [ g4^2 ] with orders [ 2 ],[128X[104X
    [4X[28X  Pcp [ g4^4 ] with orders [ 2 ],[128X[104X
    [4X[28X  Pcp [ g4^8 ] with orders [ 0 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XPcpsBySeries( DerivedSeriesOfGroup(G), "snf" );[127X[104X
    [4X[28X[ Pcp [ g2, g3, g1 ] with orders [ 2, 2, 4 ],[128X[104X
    [4X[28X  Pcp [ g4^2, g3^-2, g2^2*g4^2 ] with orders [ 4, 0, 0 ],[128X[104X
    [4X[28X  Pcp [ g4^8 ] with orders [ 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XG.1^4 in DerivedSubgroup( G );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XG.1^2 = G.4;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27X PcpsOfEfaSeries( G );[127X[104X
    [4X[28X[ Pcp [ g1 ] with orders [ 2 ],[128X[104X
    [4X[28X  Pcp [ g2 ] with orders [ 0 ],[128X[104X
    [4X[28X  Pcp [ g3 ] with orders [ 0 ],[128X[104X
    [4X[28X  Pcp [ g4 ] with orders [ 0 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X7.2 [33X[0;0YOrbit stabilizer methods for pcp-groups[133X[101X
  
  [33X[0;0YLet  [3XU[103X be a pcp-group which acts on a set [22XΩ[122X. One of the fundamental problems
  in  algorithmic  group theory is the determination of orbits and stabilizers
  of  points  in  [22XΩ[122X  under the action of [3XU[103X. We distinguish two cases: the case
  that  all  considered orbits are finite and the case that there are infinite
  orbits.  In  the latter case, an orbit cannot be listed and a description of
  the orbit and its corresponding stabilizer is much harder to obtain.[133X
  
  [33X[0;0YIf the considered orbits are finite, then the following two functions can be
  applied   to   compute   the   considered  orbits  and  their  corresponding
  stabilizers.[133X
  
  [1X7.2-1 PcpOrbitStabilizer[101X
  
  [33X[1;0Y[29X[2XPcpOrbitStabilizer[102X( [3Xpoint[103X, [3Xgens[103X, [3Xacts[103X, [3Xoper[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPcpOrbitsStabilizers[102X( [3Xpoints[103X, [3Xgens[103X, [3Xacts[103X, [3Xoper[103X ) [32X function[133X
  
  [33X[0;0YThe  input [3Xgens[103X can be an igs or a pcp of a pcp-group [3XU[103X. The elements in the
  list  [3Xgens[103X act as the elements in the list [3Xacts[103X via the function [3Xoper[103X on the
  given points; that is, [3Xoper( point, acts[i] )[103X applies the [22Xi[122Xth generator to a
  given  point.  Thus the group defined by [3Xacts[103X must be a homomorphic image of
  the  group  defined  by [3Xgens[103X. The first function returns a record containing
  the  orbit  as component 'orbit' and and igs for the stabilizer as component
  'stab'.  The second function returns a list of records, each record contains
  'repr' and 'stab'. Both of these functions run forever on infinite orbits.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := DihedralPcpGroup( 0 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xmats := [ [[-1,0],[0,1]], [[1,1],[0,1]] ];;[127X[104X
    [4X[25Xgap>[125X [27Xpcp := Pcp(G);[127X[104X
    [4X[28XPcp [ g1, g2 ] with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27XPcpOrbitStabilizer( [0,1], pcp, mats, OnRight );[127X[104X
    [4X[28Xrec( orbit := [ [ 0, 1 ] ],[128X[104X
    [4X[28X     stab := [ g1, g2 ],[128X[104X
    [4X[28X     word := [ [ [ 1, 1 ] ], [ [ 2, 1 ] ] ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf the considered orbits are infinite, then it may not always be possible to
  determine  a  description  of  the orbits and their stabilizers. However, as
  shown  in  [EO02]  and  [Eic02], it is possible to determine stabilizers and
  check if two elements are contained in the same orbit if the given action of
  the  polycyclic  group  is a unimodular linear action on a vector space. The
  following functions are available for this case.[133X
  
  [1X7.2-2 StabilizerIntegralAction[101X
  
  [33X[1;0Y[29X[2XStabilizerIntegralAction[102X( [3XU[103X, [3Xmats[103X, [3Xv[103X ) [32X function[133X
  [33X[1;0Y[29X[2XOrbitIntegralAction[102X( [3XU[103X, [3Xmats[103X, [3Xv[103X, [3Xw[103X ) [32X function[133X
  
  [33X[0;0YThe  first  function  computes the stabilizer in [3XU[103X of the vector [3Xv[103X where the
  pcp  group  [3XU[103X acts via [3Xmats[103X on an integral space and [3Xv[103X and [3Xw[103X are elements in
  this  integral  space. The second function checks whether [3Xv[103X and [3Xw[103X are in the
  same  orbit  and the function returns either [3Xfalse[103X or a record containing an
  element in [3XU[103X mapping [3Xv[103X to [3Xw[103X and the stabilizer of [3Xv[103X.[133X
  
  [1X7.2-3 NormalizerIntegralAction[101X
  
  [33X[1;0Y[29X[2XNormalizerIntegralAction[102X( [3XU[103X, [3Xmats[103X, [3XB[103X ) [32X function[133X
  [33X[1;0Y[29X[2XConjugacyIntegralAction[102X( [3XU[103X, [3Xmats[103X, [3XB[103X, [3XC[103X ) [32X function[133X
  
  [33X[0;0YThe  first  function  computes  the  normalizer in [3XU[103X of the lattice with the
  basis [3XB[103X, where the pcp group [3XU[103X acts via [3Xmats[103X on an integral space and [3XB[103X is a
  subspace of this integral space. The second functions checks whether the two
  lattices with the bases [3XB[103X and [3XC[103X are contained in the same orbit under [3XU[103X. The
  function  returns either [3Xfalse[103X or a record with an element in [3XU[103X mapping [3XB[103X to
  [3XC[103X and the stabilizer of [3XB[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X# get a pcp group and a free abelian normal subgroup[128X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfSomePcpGroups(8);[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xefa := EfaSeries(G);[127X[104X
    [4X[28X[ Pcp-group with orders [ 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X  Pcp-group with orders [ 0, 0, 0, 0 ],[128X[104X
    [4X[28X  Pcp-group with orders [ 0, 0, 0 ],[128X[104X
    [4X[28X  Pcp-group with orders [  ] ][128X[104X
    [4X[25Xgap>[125X [27XN := efa[3];[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XIsFreeAbelian(N);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
    [4X[28X# create conjugation action on N[128X[104X
    [4X[25Xgap>[125X [27Xmats := LinearActionOnPcp(Igs(G), Pcp(N));[127X[104X
    [4X[28X[ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],[128X[104X
    [4X[28X  [ [ 0, 0, 1 ], [ 1, -1, 1 ], [ 0, 1, 0 ] ],[128X[104X
    [4X[28X  [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],[128X[104X
    [4X[28X  [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],[128X[104X
    [4X[28X  [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28X# take an arbitrary vector and compute its stabilizer[128X[104X
    [4X[25Xgap>[125X [27XStabilizerIntegralAction(G,mats, [2,3,4]);[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XIgs(last);[127X[104X
    [4X[28X[ g1, g3, g4, g5 ][128X[104X
    [4X[28X[128X[104X
    [4X[28X# check orbits with some other vectors[128X[104X
    [4X[25Xgap>[125X [27XOrbitIntegralAction(G,mats, [2,3,4],[3,1,5]);[127X[104X
    [4X[28Xrec( stab := Pcp-group with orders [ 0, 0, 0, 0 ], prei := g2 )[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XOrbitIntegralAction(G,mats, [2,3,4], [4,6,8]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
    [4X[28X# compute the orbit of a subgroup of Z^3 under the action of G[128X[104X
    [4X[25Xgap>[125X [27XNormalizerIntegralAction(G, mats, [[1,0,0],[0,1,0]]);[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XIgs(last);[127X[104X
    [4X[28X[ g1, g2^2, g3, g4, g5 ][128X[104X
  [4X[32X[104X
  
  
  [1X7.3 [33X[0;0YCentralizers, Normalizers and Intersections[133X[101X
  
  [33X[0;0YIn  this  section we list a number of operations for which there are methods
  installed to compute the corresponding features in polycyclic groups.[133X
  
  [1X7.3-1 Centralizer[101X
  
  [33X[1;0Y[29X[2XCentralizer[102X( [3XU[103X, [3Xg[103X ) [32X method[133X
  [33X[1;0Y[29X[2XIsConjugate[102X( [3XU[103X, [3Xg[103X, [3Xh[103X ) [32X method[133X
  
  [33X[0;0YThese  functions  solve the conjugacy problem for elements in pcp-groups and
  they  can  be  used  to  compute  centralizers.  The  first method returns a
  subgroup  of  the  given  group  [3XU[103X,  the  second  method  either  returns  a
  conjugating element or false if no such element exists.[133X
  
  [33X[0;0YThe  methods  are  based  on  the  orbit  stabilizer algorithms described in
  [EO02].  For  nilpotent  groups, an algorithm to solve the conjugacy problem
  for elements is described in [Sim94].[133X
  
  [1X7.3-2 Centralizer[101X
  
  [33X[1;0Y[29X[2XCentralizer[102X( [3XU[103X, [3XV[103X ) [32X method[133X
  [33X[1;0Y[29X[2XNormalizer[102X( [3XU[103X, [3XV[103X ) [32X method[133X
  [33X[1;0Y[29X[2XIsConjugate[102X( [3XU[103X, [3XV[103X, [3XW[103X ) [32X method[133X
  
  [33X[0;0YThese  three functions solve the conjugacy problem for subgroups and compute
  centralizers  and  normalizers  of subgroups. The first two functions return
  subgroups  of  the  input  group [3XU[103X, the third function returns a conjugating
  element or false if no such element exists.[133X
  
  [33X[0;0YThe  methods  are  based  on  the  orbit  stabilizer algorithms described in
  [Eic02].  For nilpotent groups, an algorithm to solve the conjugacy problems
  for subgroups is described in [Lo98b].[133X
  
  [1X7.3-3 Intersection[101X
  
  [33X[1;0Y[29X[2XIntersection[102X( [3XU[103X, [3XN[103X ) [32X function[133X
  
  [33X[0;0YA  general  method  to  compute intersections of subgroups of a pcp-group is
  described  in  [Eic01a],  but  it  is  not  yet  implemented  here. However,
  intersections  of  subgroups [22XU, N ≤ G[122X can be computed if [22XN[122X is normalising [22XU[122X.
  See [Sim94] for an outline of the algorithm.[133X
  
  
  [1X7.4 [33X[0;0YFinite subgroups[133X[101X
  
  [33X[0;0YThere  are  various  finite  subgroups of interest in polycyclic groups. See
  [Eic00] for a description of the algorithms underlying the functions in this
  section.[133X
  
  [1X7.4-1 TorsionSubgroup[101X
  
  [33X[1;0Y[29X[2XTorsionSubgroup[102X( [3XU[103X ) [32X attribute[133X
  
  [33X[0;0YIf the set of elements of finite order forms a subgroup, then we call it the
  [13Xtorsion  subgroup[113X. This function determines the torsion subgroup of [3XU[103X, if it
  exists, and returns fail otherwise. Note that a torsion subgroup does always
  exist if [3XU[103X is nilpotent.[133X
  
  [1X7.4-2 NormalTorsionSubgroup[101X
  
  [33X[1;0Y[29X[2XNormalTorsionSubgroup[102X( [3XU[103X ) [32X attribute[133X
  
  [33X[0;0YEach  polycyclic  groups  has  a unique largest finite normal subgroup. This
  function computes it for [3XU[103X.[133X
  
  [1X7.4-3 IsTorsionFree[101X
  
  [33X[1;0Y[29X[2XIsTorsionFree[102X( [3XU[103X ) [32X property[133X
  
  [33X[0;0YThis function checks if [3XU[103X is torsion free. It returns true or false.[133X
  
  [1X7.4-4 FiniteSubgroupClasses[101X
  
  [33X[1;0Y[29X[2XFiniteSubgroupClasses[102X( [3XU[103X ) [32X attribute[133X
  
  [33X[0;0YThere  exist  only  finitely many conjugacy classes of finite subgroups in a
  polycyclic  group  [3XU[103X  and  this  function  can  be used to compute them. The
  algorithm  underlying this function proceeds by working down a normal series
  of  [3XU[103X with elementary or free abelian factors. The following function can be
  used to give the algorithm a specific series.[133X
  
  [1X7.4-5 FiniteSubgroupClassesBySeries[101X
  
  [33X[1;0Y[29X[2XFiniteSubgroupClassesBySeries[102X( [3XU[103X, [3Xpcps[103X ) [32X function[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfSomePcpGroups(15);[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 0 ][128X[104X
    [4X[25Xgap>[125X [27XTorsionSubgroup(G);[127X[104X
    [4X[28XPcp-group with orders [ 5, 2 ][128X[104X
    [4X[25Xgap>[125X [27XNormalTorsionSubgroup(G);[127X[104X
    [4X[28XPcp-group with orders [ 5, 2 ][128X[104X
    [4X[25Xgap>[125X [27XIsTorsionFree(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XFiniteSubgroupClasses(G);[127X[104X
    [4X[28X[ Pcp-group with orders [ 5, 2 ]^G,[128X[104X
    [4X[28X  Pcp-group with orders [ 2 ]^G,[128X[104X
    [4X[28X  Pcp-group with orders [ 5 ]^G,[128X[104X
    [4X[28X  Pcp-group with orders [  ]^G ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG := DihedralPcpGroup( 0 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27XTorsionSubgroup(G);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XNormalTorsionSubgroup(G);[127X[104X
    [4X[28XPcp-group with orders [  ][128X[104X
    [4X[25Xgap>[125X [27XIsTorsionFree(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XFiniteSubgroupClasses(G);[127X[104X
    [4X[28X[ Pcp-group with orders [ 2 ]^G,[128X[104X
    [4X[28X  Pcp-group with orders [ 2 ]^G,[128X[104X
    [4X[28X  Pcp-group with orders [  ]^G ][128X[104X
  [4X[32X[104X
  
  
  [1X7.5 [33X[0;0YSubgroups of finite index and maximal subgroups[133X[101X
  
  [33X[0;0YHere  we outline functions to determine various types of subgroups of finite
  index  in  polycyclic  groups.  Again,  see [Eic00] for a description of the
  algorithms  underlying  the  functions  in  this  section. Also, we refer to
  [Lo98a] for an alternative approach.[133X
  
  [1X7.5-1 MaximalSubgroupClassesByIndex[101X
  
  [33X[1;0Y[29X[2XMaximalSubgroupClassesByIndex[102X( [3XU[103X, [3Xp[103X ) [32X operation[133X
  
  [33X[0;0YEach  maximal  subgroup  of  a polycyclic group [3XU[103X has [3Xp[103X-power index for some
  prime [3Xp[103X. This function can be used to determine the conjugacy classes of all
  maximal subgroups of [3Xp[103X-power index for a given prime [3Xp[103X.[133X
  
  [1X7.5-2 LowIndexSubgroupClasses[101X
  
  [33X[1;0Y[29X[2XLowIndexSubgroupClasses[102X( [3XU[103X, [3Xn[103X ) [32X operation[133X
  
  [33X[0;0YThere  are  only  finitely  many  subgroups of a given index in a polycyclic
  group  [3XU[103X. This function computes conjugacy classes of all subgroups of index
  [3Xn[103X in [3XU[103X.[133X
  
  [1X7.5-3 LowIndexNormalSubgroups[101X
  
  [33X[1;0Y[29X[2XLowIndexNormalSubgroups[102X( [3XU[103X, [3Xn[103X ) [32X operation[133X
  
  [33X[0;0YThis function computes the normal subgroups of index [3Xn[103X in [3XU[103X.[133X
  
  [1X7.5-4 NilpotentByAbelianNormalSubgroup[101X
  
  [33X[1;0Y[29X[2XNilpotentByAbelianNormalSubgroup[102X( [3XU[103X ) [32X function[133X
  
  [33X[0;0YThis  function  returns a normal subgroup [3XN[103X of finite index in [3XU[103X such that [3XN[103X
  is  nilpotent-by-abelian.  Such  a subgroup exists in every polycyclic group
  and this function computes such a subgroup using LowIndexNormal. However, we
  note   that   this   function   is  not  very  efficient  and  the  function
  NilpotentByAbelianByFiniteSeries may well be more efficient on this task.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfSomePcpGroups(2);[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XMaximalSubgroupClassesByIndex( G, 61 );;[127X[104X
    [4X[25Xgap>[125X [27Xmax := List( last, Representative );;[127X[104X
    [4X[25Xgap>[125X [27XList( max, x -> Index( G, x ) );[127X[104X
    [4X[28X[ 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,[128X[104X
    [4X[28X  61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,[128X[104X
    [4X[28X  61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,[128X[104X
    [4X[28X  61, 61, 61, 61, 61, 61, 226981 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XLowIndexSubgroupClasses( G, 61 );;[127X[104X
    [4X[25Xgap>[125X [27Xlow := List( last, Representative );;[127X[104X
    [4X[25Xgap>[125X [27XList( low, x -> Index( G, x ) );[127X[104X
    [4X[28X[ 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,[128X[104X
    [4X[28X  61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,[128X[104X
    [4X[28X  61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,[128X[104X
    [4X[28X  61, 61, 61, 61, 61, 61 ][128X[104X
  [4X[32X[104X
  
  
  [1X7.6 [33X[0;0YFurther attributes for pcp-groups based on the Fitting subgroup[133X[101X
  
  [33X[0;0YIn  this  section  we  provide a variety of other attributes for pcp-groups.
  Most  of  the  methods below are based or related to the Fitting subgroup of
  the  given  group.  We refer to [Eic01b] for a description of the underlying
  methods.[133X
  
  [1X7.6-1 FittingSubgroup[101X
  
  [33X[1;0Y[29X[2XFittingSubgroup[102X( [3XU[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  the  Fitting  subgroup  of [3XU[103X; that is, the largest nilpotent normal
  subgroup of [3XU[103X.[133X
  
  [1X7.6-2 IsNilpotentByFinite[101X
  
  [33X[1;0Y[29X[2XIsNilpotentByFinite[102X( [3XU[103X ) [32X property[133X
  
  [33X[0;0Ychecks whether the Fitting subgroup of [3XU[103X has finite index.[133X
  
  [1X7.6-3 Centre[101X
  
  [33X[1;0Y[29X[2XCentre[102X( [3XU[103X ) [32X method[133X
  
  [33X[0;0Yreturns the centre of [3XU[103X.[133X
  
  [1X7.6-4 FCCentre[101X
  
  [33X[1;0Y[29X[2XFCCentre[102X( [3XU[103X ) [32X method[133X
  
  [33X[0;0Yreturns  the  FC-centre  of [3XU[103X; that is, the subgroup containing all elements
  having a finite conjugacy class in [3XU[103X.[133X
  
  [1X7.6-5 PolyZNormalSubgroup[101X
  
  [33X[1;0Y[29X[2XPolyZNormalSubgroup[102X( [3XU[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  normal  subgroup  [3XN[103X  of  finite  index  in  [3XU[103X, such that [3XN[103X has a
  polycyclic series with infinite factors only.[133X
  
  [1X7.6-6 NilpotentByAbelianByFiniteSeries[101X
  
  [33X[1;0Y[29X[2XNilpotentByAbelianByFiniteSeries[102X( [3XU[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  normal  series  [22X1  ≤  F ≤ A ≤ U[122X such that [22XF[122X is nilpotent, [22XA/F[122X is
  abelian  and  [22XU/A[122X  is  finite.  This  series  is  computed using the Fitting
  subgroup and the centre of the Fitting factor.[133X
  
  
  [1X7.7 [33X[0;0YFunctions for nilpotent groups[133X[101X
  
  [33X[0;0YThere  are  (very  few)  functions  which are available for nilpotent groups
  only. First, there are the different central series. These are available for
  all  groups,  but  for  nilpotent  groups  they terminate and provide series
  through  the full group. Secondly, the determination of a minimal generating
  set is available for nilpotent groups only.[133X
  
  [1X7.7-1 MinimalGeneratingSet[101X
  
  [33X[1;0Y[29X[2XMinimalGeneratingSet[102X( [3XU[103X ) [32X method[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfSomePcpGroups(14);[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 0, 5, 5, 4, 0, 6,[128X[104X
    [4X[28X  5, 5, 4, 0, 10, 6 ][128X[104X
    [4X[25Xgap>[125X [27XIsNilpotent(G);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XPcpsBySeries( LowerCentralSeriesOfGroup(G));[127X[104X
    [4X[28X[ Pcp [ g1, g2 ] with orders [ 0, 0 ],[128X[104X
    [4X[28X  Pcp [ g3 ] with orders [ 0 ],[128X[104X
    [4X[28X  Pcp [ g4 ] with orders [ 0 ],[128X[104X
    [4X[28X  Pcp [ g5 ] with orders [ 0 ],[128X[104X
    [4X[28X  Pcp [ g6, g7 ] with orders [ 0, 0 ],[128X[104X
    [4X[28X  Pcp [ g8 ] with orders [ 0 ],[128X[104X
    [4X[28X  Pcp [ g9, g10 ] with orders [ 0, 0 ],[128X[104X
    [4X[28X  Pcp [ g11, g12, g13 ] with orders [ 5, 4, 0 ],[128X[104X
    [4X[28X  Pcp [ g14, g15, g16, g17, g18 ] with orders [ 5, 5, 4, 0, 6 ],[128X[104X
    [4X[28X  Pcp [ g19, g20, g21, g22, g23, g24 ] with orders [ 5, 5, 4, 0, 10, 6 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XPcpsBySeries( UpperCentralSeriesOfGroup(G));[127X[104X
    [4X[28X[ Pcp [ g1, g2 ] with orders [ 0, 0 ],[128X[104X
    [4X[28X  Pcp [ g3 ] with orders [ 0 ],[128X[104X
    [4X[28X  Pcp [ g4 ] with orders [ 0 ],[128X[104X
    [4X[28X  Pcp [ g5 ] with orders [ 0 ],[128X[104X
    [4X[28X  Pcp [ g6, g7 ] with orders [ 0, 0 ],[128X[104X
    [4X[28X  Pcp [ g8 ] with orders [ 0 ],[128X[104X
    [4X[28X  Pcp [ g9, g10 ] with orders [ 0, 0 ],[128X[104X
    [4X[28X  Pcp [ g11, g12, g13 ] with orders [ 5, 4, 0 ],[128X[104X
    [4X[28X  Pcp [ g14, g15, g16, g17, g18 ] with orders [ 5, 5, 4, 0, 6 ],[128X[104X
    [4X[28X  Pcp [ g19, g20, g21, g22, g23, g24 ] with orders [ 5, 5, 4, 0, 10, 6 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XMinimalGeneratingSet(G);[127X[104X
    [4X[28X[ g1, g2 ][128X[104X
  [4X[32X[104X
  
  
  [1X7.8 [33X[0;0YRandom methods for pcp-groups[133X[101X
  
  [33X[0;0YBelow  we  introduce  a function which computes orbit and stabilizer using a
  random  method.  This  function  tries  to  approximate  the  orbit  and the
  stabilizer,  but  the  returned  orbit or stabilizer may be incomplete. This
  function   is  used  in  the  random  methods  to  compute  normalizers  and
  centralizers.  Note  that  deterministic methods for these purposes are also
  available.[133X
  
  [1X7.8-1 RandomCentralizerPcpGroup[101X
  
  [33X[1;0Y[29X[2XRandomCentralizerPcpGroup[102X( [3XU[103X, [3Xg[103X ) [32X function[133X
  [33X[1;0Y[29X[2XRandomCentralizerPcpGroup[102X( [3XU[103X, [3XV[103X ) [32X function[133X
  [33X[1;0Y[29X[2XRandomNormalizerPcpGroup[102X( [3XU[103X, [3XV[103X ) [32X function[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := DihedralPcpGroup(0);[127X[104X
    [4X[28XPcp-group with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xmats := [[[-1, 0],[0,1]], [[1,1],[0,1]]];[127X[104X
    [4X[28X[ [ [ -1, 0 ], [ 0, 1 ] ], [ [ 1, 1 ], [ 0, 1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xpcp := Pcp(G);[127X[104X
    [4X[28XPcp [ g1, g2 ] with orders [ 2, 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XRandomPcpOrbitStabilizer( [1,0], pcp, mats, OnRight ).stab;[127X[104X
    [4X[28X#I  Orbit longer than limit: exiting.[128X[104X
    [4X[28X[  ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xg := Igs(G)[1];[127X[104X
    [4X[28Xg1[128X[104X
    [4X[25Xgap>[125X [27XRandomCentralizerPcpGroup( G, g );[127X[104X
    [4X[28X#I  Stabilizer not increasing: exiting.[128X[104X
    [4X[28XPcp-group with orders [ 2 ][128X[104X
    [4X[25Xgap>[125X [27XIgs(last);[127X[104X
    [4X[28X[ g1 ][128X[104X
  [4X[32X[104X
  
  
  [1X7.9 [33X[0;0YNon-abelian tensor product and Schur extensions[133X[101X
  
  [1X7.9-1 SchurExtension[101X
  
  [33X[1;0Y[29X[2XSchurExtension[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YLet [3XG[103X be a polycyclic group with a polycyclic generating sequence consisting
  of  [22Xn[122X  elements. This function computes the largest central extension [3XH[103X of [3XG[103X
  such  that [3XH[103X is generated by [22Xn[122X elements. If [22XF/R[122X is the underlying polycyclic
  presentation for [3XG[103X, then [3XH[103X is isomorphic to [22XF/[R,F][122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := DihedralPcpGroup( 0 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27XCentre( G );[127X[104X
    [4X[28XPcp-group with orders [  ][128X[104X
    [4X[25Xgap>[125X [27XH := SchurExtension( G );[127X[104X
    [4X[28XPcp-group with orders [ 2, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XCentre( H );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XH/Centre(H);[127X[104X
    [4X[28XPcp-group with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27XSubgroup( H, [H.1,H.2] ) = H;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.9-2 SchurExtensionEpimorphism[101X
  
  [33X[1;0Y[29X[2XSchurExtensionEpimorphism[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  the  projection  from  the Schur extension [22XG^*[122X of [3XG[103X onto [3XG[103X. See the
  function  [10XSchurExtension[110X.  The  kernel  of  this  epimorphism  is the direct
  product  of the Schur multiplicator of [3XG[103X and a direct product of [22Xn[122X copies of
  [22Xℤ[122X  where [22Xn[122X is the number of generators in the polycyclic presentation for [3XG[103X.
  The  Schur  multiplicator  is the intersection of the kernel and the derived
  group of the source. See also the function [10XSchurCover[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgl23 := Range( IsomorphismPcpGroup( GL(2,3) ) );[127X[104X
    [4X[28XPcp-group with orders [ 2, 3, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XSchurExtensionEpimorphism( gl23 );[127X[104X
    [4X[28X[ g1, g2, g3, g4, g5, g6, g7, g8, g9, g10 ] -> [ g1, g2, g3, g4, g5,[128X[104X
    [4X[28Xid, id, id, id, id ][128X[104X
    [4X[25Xgap>[125X [27XKernel( last );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XAbelianInvariantsMultiplier( gl23 );[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XIntersection( Kernel(epi), DerivedSubgroup( Source(epi) ) );[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere  is a crossed pairing from [3XG[103X into [22X(G^*)'[122X which can be defined via this
  epimorphism:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := DihedralPcpGroup(0);[127X[104X
    [4X[28XPcp-group with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xepi := SchurExtensionEpimorphism( G );[127X[104X
    [4X[28X[ g1, g2, g3, g4 ] -> [ g1, g2, id, id ][128X[104X
    [4X[25Xgap>[125X [27XPreImagesRepresentative( epi, G.1 );[127X[104X
    [4X[28Xg1[128X[104X
    [4X[25Xgap>[125X [27XPreImagesRepresentative( epi, G.2 );[127X[104X
    [4X[28Xg2[128X[104X
    [4X[25Xgap>[125X [27XComm( last, last2 );[127X[104X
    [4X[28Xg2^-2*g4[128X[104X
  [4X[32X[104X
  
  [1X7.9-3 SchurCover[101X
  
  [33X[1;0Y[29X[2XSchurCover[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0Ycomputes  a Schur covering group of the polycyclic group [3XG[103X. A Schur covering
  is  a  largest  central  extension  [3XH[103X  of  [3XG[103X  such  that the kernel [3XM[103X of the
  projection of [3XH[103X onto [3XG[103X is contained in the commutator subgroup of [3XH[103X.[133X
  
  [33X[0;0YIf  [3XG[103X is given by a presentation [22XF/R[122X, then [3XM[103X is isomorphic to the subgroup [22XR
  ∩ [F,F] / [R,F][122X. Let [22XC[122X be a complement to [22XR ∩ [F,F] / [R,F][122X in [22XR/[R,F][122X. Then
  [22XF/C[122X is isomorphic to [3XH[103X and [22XR/C[122X is isomorphic to [3XM[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := AbelianPcpGroup( 3 );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xext := SchurCover( G );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XCentre( ext );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XIsSubgroup( DerivedSubgroup( ext ), last );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.9-4 AbelianInvariantsMultiplier[101X
  
  [33X[1;0Y[29X[2XAbelianInvariantsMultiplier[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns a list of the abelian invariants of the Schur multiplier of G.[133X
  
  [33X[0;0YNote  that  the  Schur  multiplicator  of  a  polycyclic group is a finitely
  generated abelian group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := DihedralPcpGroup( 0 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27XDirectProduct( G, AbelianPcpGroup( 2 ) );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27XAbelianInvariantsMultiplier( last );[127X[104X
    [4X[28X[ 0, 2, 2, 2, 2 ][128X[104X
  [4X[32X[104X
  
  [1X7.9-5 NonAbelianExteriorSquareEpimorphism[101X
  
  [33X[1;0Y[29X[2XNonAbelianExteriorSquareEpimorphism[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the  epimorphism of the non-abelian exterior square of a polycyclic
  group  [3XG[103X onto the derived group of [3XG[103X. The non-abelian exterior square can be
  defined  as the derived subgroup of a Schur cover of [3XG[103X. The isomorphism type
  of  the  non-abelian  exterior  square  is  unique despite the fact that the
  isomorphism type of a Schur cover of a polycyclic groups need not be unique.
  The derived group of a Schur cover has a natural projection onto the derived
  group of [3XG[103X which is what the function returns.[133X
  
  [33X[0;0YThe kernel of the epimorphism is isomorphic to the Schur multiplicator of [3XG[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfSomePcpGroups( 3 );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XG := DirectProduct( G,G );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XAbelianInvariantsMultiplier( G );[127X[104X
    [4X[28X[ [ 0, 1 ], [ 2, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xepi := NonAbelianExteriorSquareEpimorphism( G );[127X[104X
    [4X[28X[ g2^-2*g5, g4^-2*g10, g6, g7, g8, g9 ] -> [ g2^-2, g4^-2, id, id, id, id ][128X[104X
    [4X[25Xgap>[125X [27XKernel( epi );[127X[104X
    [4X[28XPcp-group with orders [ 0, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XCollected( AbelianInvariants( last ) );[127X[104X
    [4X[28X[ [ 0, 1 ], [ 2, 3 ] ][128X[104X
  [4X[32X[104X
  
  [1X7.9-6 NonAbelianExteriorSquare[101X
  
  [33X[1;0Y[29X[2XNonAbelianExteriorSquare[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0Ycomputes  the  non-abelian  exterior square of a polycyclic group [3XG[103X. See the
  explanation  for [10XNonAbelianExteriorSquareEpimorphism[110X. The natural projection
  of  the non-abelian exterior square onto the derived group of [3XG[103X is stored in
  the component [10X!.epimorphism[110X.[133X
  
  [33X[0;0YThere  is  a  crossed  pairing  from  [22XG×  G[122X  into  [22XG∧  G[122X.  See  the function
  [10XSchurExtensionEpimorphism[110X  for details. The crossed pairing is stored in the
  component [10X!.crossedPairing[110X. This is the crossed pairing [22Xλ[122X in [EN08].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := DihedralPcpGroup(0);[127X[104X
    [4X[28XPcp-group with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27XGwG := NonAbelianExteriorSquare( G );[127X[104X
    [4X[28XPcp-group with orders [ 0 ][128X[104X
    [4X[25Xgap>[125X [27Xlambda := GwG!.crossedPairing;[127X[104X
    [4X[28Xfunction( g, h ) ... end[128X[104X
    [4X[25Xgap>[125X [27Xlambda( G.1, G.2 );[127X[104X
    [4X[28Xg2^2*g4^-1[128X[104X
  [4X[32X[104X
  
  [1X7.9-7 NonAbelianTensorSquareEpimorphism[101X
  
  [33X[1;0Y[29X[2XNonAbelianTensorSquareEpimorphism[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0Yreturns  for  a  polycyclic group [3XG[103X the projection of the non-abelian tensor
  square  [22XG⊗  G[122X  onto  the non-abelian exterior square [22XG∧ G[122X. The range of that
  epimorphism  has  the  component  [10X!.epimorphism[110X set to the projection of the
  non-abelian  exterior  square  onto  the  derived  group  of [3XG[103X. See also the
  function [10XNonAbelianExteriorSquare[110X.[133X
  
  [33X[0;0YWith  the  result  of  this  function  one  can  compute  the  groups in the
  commutative  diagram at the beginning of the paper [EN08]. The kernel of the
  returned  epimorphism  is  the  group [22X∇(G)[122X. The kernel of the composition of
  this epimorphism and the above mention projection onto [22XG'[122X is the group [22XJ(G)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := DihedralPcpGroup(0);[127X[104X
    [4X[28XPcp-group with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27XG := DirectProduct(G,G);[127X[104X
    [4X[28XPcp-group with orders [ 2, 0, 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xalpha := NonAbelianTensorSquareEpimorphism( G );[127X[104X
    [4X[28X[ g9*g25^-1, g10*g26^-1, g11*g27, g12*g28, g13*g29, g14*g30, g15, g16,[128X[104X
    [4X[28Xg17,[128X[104X
    [4X[28X  g18, g19, g20, g21, g22, g23, g24 ] -> [ g2^-2*g6, g4^-2*g12, g8,[128X[104X
    [4X[28X  g9, g10,[128X[104X
    [4X[28X  g11, id, id, id, id, id, id, id, id, id, id ][128X[104X
    [4X[25Xgap>[125X [27Xgamma := Range( alpha )!.epimorphism;[127X[104X
    [4X[28X[ g2^-2*g6, g4^-2*g12, g8, g9, g10, g11 ] -> [ g2^-2, g4^-2, id, id,[128X[104X
    [4X[28Xid, id ][128X[104X
    [4X[25Xgap>[125X [27XJG := Kernel( alpha * gamma );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XImage( alpha, JG );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XAbelianInvariantsMultiplier( G );[127X[104X
    [4X[28X[ [ 2, 4 ] ][128X[104X
  [4X[32X[104X
  
  [1X7.9-8 NonAbelianTensorSquare[101X
  
  [33X[1;0Y[29X[2XNonAbelianTensorSquare[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0Ycomputes for a polycyclic group [3XG[103X the non-abelian tensor square [22XG⊗ G[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := AlternatingGroup( IsPcGroup, 4 );[127X[104X
    [4X[28X<pc group of size 12 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XPcGroupToPcpGroup( G );[127X[104X
    [4X[28XPcp-group with orders [ 3, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XNonAbelianTensorSquare( last );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XPcpGroupToPcGroup( last );[127X[104X
    [4X[28X<pc group of size 24 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XDirectFactorsOfGroup( last );[127X[104X
    [4X[28X[ Group([ f1, f2, f3 ]), Group([ f4 ]) ][128X[104X
    [4X[25Xgap>[125X [27XList( last, Size );[127X[104X
    [4X[28X[ 8, 3 ][128X[104X
    [4X[25Xgap>[125X [27XIdGroup( last2[1] );[127X[104X
    [4X[28X[ 8, 4 ]       # the quaternion group of Order 8[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG := DihedralPcpGroup( 0 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xten := NonAbelianTensorSquare( G );[127X[104X
    [4X[28XPcp-group with orders [ 0, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XIsAbelian( ten );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.9-9 NonAbelianExteriorSquarePlusEmbedding[101X
  
  [33X[1;0Y[29X[2XNonAbelianExteriorSquarePlusEmbedding[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0Yreturns  an  embedding  from  the  non-abelian  exterior square [22XG∧ G[122X into an
  extensions  of [22XG∧ G[122X by [22XG× G[122X. For the significance of the group see the paper
  [EN08]. The range of the epimorphism is the group [22Xτ(G)[122X in that paper.[133X
  
  [1X7.9-10 NonAbelianTensorSquarePlusEpimorphism[101X
  
  [33X[1;0Y[29X[2XNonAbelianTensorSquarePlusEpimorphism[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0Yreturns an epimorphisms of [22Xν(G)[122X onto [22Xτ(G)[122X. The group [22Xν(G)[122X is an extension of
  the  non-abelian  tensor  square  [22XG⊗  G[122X  of  [22XG[122X by [22XG× G[122X. The group [22Xτ(G)[122X is an
  extension  of  the non-abelian exterior square [22XG∧ G[122X by [22XG× G[122X. For details see
  [EN08].[133X
  
  [1X7.9-11 NonAbelianTensorSquarePlus[101X
  
  [33X[1;0Y[29X[2XNonAbelianTensorSquarePlus[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0Yreturns the group [22Xν(G)[122X in [EN08].[133X
  
  [1X7.9-12 WhiteheadQuadraticFunctor[101X
  
  [33X[1;0Y[29X[2XWhiteheadQuadraticFunctor[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0Yreturns  Whitehead's  universal  quadratic  functor  of  [22XG[122X, see [EN08] for a
  description.[133X
  
  
  [1X7.10 [33X[0;0YSchur covers[133X[101X
  
  [33X[0;0YThis  section  contains a function to determine the Schur covers of a finite
  [22Xp[122X-group up to isomorphism.[133X
  
  [1X7.10-1 SchurCovers[101X
  
  [33X[1;0Y[29X[2XSchurCovers[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0YLet  [3XG[103X  be  a finite [22Xp[122X-group defined as a pcp group. This function returns a
  complete  and irredundant set of isomorphism types of Schur covers of [3XG[103X. The
  algorithm implements a method of Nickel's Phd Thesis.[133X
  
