  
  [1X1 [33X[0;0YThe Perfect Groups Library[133X[101X
  
  [33X[0;0YThe [5XGAP[105X library of finite perfect groups provides, up to isomorphism, a list
  of  all  perfect  groups  whose  sizes  are less than [22X2⋅ 10^6[122X. The groups of
  orders  up to [22X10^6[122X have been enumerated by Derek F. Holt and Wilhelm Plesken
  and  published  in  their  book [21XPerfect Groups[121X [HP89]. For orders [22Xn = 86016[122X,
  368640,  or 737280 this work only counted the groups (but did not explicitly
  list  them), the groups of orders [22Xn = 61440[122X, 122880, 172032, 245760, 344064,
  491520, 688128, or 983040 were omitted.[133X
  
  [33X[0;0YWe  are  grateful  to Derek Holt and Wilhelm Plesken for making their groups
  available  to  the  [5XGAP[105X  community by contributing their files. It should be
  noted  that their book contains a lot of further information for many of the
  library  groups.  So  we  would  like to recommend it to any [5XGAP[105X user who is
  interested  in  the  groups.  The library of these has been brought into [5XGAP[105X
  format by Volkmar Felsch.[133X
  
  [33X[0;0YSeveral  additional  groups  omitted  from the book [21XPerfect Groups[121X have also
  been included. Two groups -- one of order 450000 with a factor group of type
  [22XA_6[122X  and  the one of order 962280 -- were found by Jack Schmidt in 2005. Two
  groups  of  order  243000 and one each of orders 729000, 871200, 878460 were
  found in 2020 by Alexander Hulpke.[133X
  
  [33X[0;0YThe  perfect  groups of size less than [22X2⋅ 10^6[122X which had not been classified
  in  the  work  of Holt and Plesken have been enumerated by Alexander Hulpke;
  this  classification is described in [Hul22], which also gives an account of
  the methods used and of the verification of the resulting data. These groups
  are  stored  directly  and  provide  less  construction information in their
  names.[133X
  
  [33X[0;0YAs  all  groups are stored by presentations, a permutation representation is
  obtained  by  coset enumeration. Note that some of the library groups do not
  have  a faithful permutation representation of small degree. Computations in
  these groups may be rather time consuming.[133X
  
  [33X[0;0YThe  perfect  subgroups  of  a  given  finite  group  can be determined with
  [2XRepresentativesPerfectSubgroups[102X ([14XReference: RepresentativesPerfectSubgroups[114X)
  and               [2XConjugacyClassesPerfectSubgroups[102X               ([14XReference:
  ConjugacyClassesPerfectSubgroups[114X), which are declared by the [5XGAP[105X library but
  require this package in order to work.[133X
  
  
  [1X1.1 [33X[0;0YAccessing the Perfect Groups Library[133X[101X
  
  [1X1.1-1 SizesPerfectGroups[101X
  
  [33X[1;0Y[29X[2XSizesPerfectGroups[102X(  ) [32X function[133X
  
  [33X[0;0YThis is the ordered list of all numbers up to [22X2⋅ 10^6[122X that occur as sizes of
  perfect  groups.  One  can  iterate  over part of the perfect groups library
  with:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfor n in Intersection([100..500],SizesPerfectGroups()) do[127X[104X
    [4X[25X>[125X [27X     for k in [1..NrPerfectGroups(n)] do[127X[104X
    [4X[25X>[125X [27X       pg := PerfectGroup(n,k);[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  
  [1X1.1-2 [33X[0;0YPerfectGroup[133X[101X
  
  [33X[1;0Y[29X[2XPerfectGroup[102X( [[3Xfilt[103X, ][3Xsize[103X[, [3Xn[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XPerfectGroup[102X( [[3Xfilt[103X, ][3Xsizenumberpair[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  group  which is isomorphic to the library group specified by the
  size  number  [10X[  [3Xsize[103X[10X,  [3Xn[103X[10X  ][110X  or  by  the two separate arguments [3Xsize[103X and [3Xn[103X,
  assuming  a  default  value of [22X[3Xn[103X = 1[122X. The optional argument [3Xfilt[103X defines the
  filter  in  which  the  group  is  returned.  Possible  filters  so  far are
  [2XIsPermGroup[102X   ([14XReference:  IsPermGroup[114X)  and  [2XIsSubgroupFpGroup[102X  ([14XReference:
  IsSubgroupFpGroup[114X).  In  the  latter  case, the generators and relators used
  coincide  with  those  given  in [HP89].  The  default filter is [2XIsPermGroup[102X
  ([14XReference: IsPermGroup[114X).[133X
  
  [33X[0;0YThe  data  for  the  orders 1376256 and 1966080 is not distributed with this
  package;  for  these,  this  function  enters a break loop explaining how to
  obtain the missing file, see [14X1.2[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := PerfectGroup(IsPermGroup,6048,1);[127X[104X
    [4X[28XU3(3)[128X[104X
    [4X[25Xgap>[125X [27XG:=PerfectGroup(IsPermGroup,823080,2);[127X[104X
    [4X[28XA5 2^1 19^2 C 19^1[128X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints(G);[127X[104X
    [4X[28X6859[128X[104X
    [4X[25Xgap>[125X [27XG:=PerfectGroup(1866240,12);[127X[104X
    [4X[28XPG1866240.12[128X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints(G);[127X[104X
    [4X[28X270[128X[104X
  [4X[32X[104X
  
  [1X1.1-3 PerfectIdentification[101X
  
  [33X[1;0Y[29X[2XPerfectIdentification[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YThis  attribute  is  set  for  all  groups  obtained from the perfect groups
  library  and  has  the  value  [10X[[3Xsize[103X[10X,[3Xnr[103X[10X][110X if the group is obtained with these
  parameters from the library.[133X
  
  [1X1.1-4 NumberPerfectGroups[101X
  
  [33X[1;0Y[29X[2XNumberPerfectGroups[102X( [3Xsize[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrPerfectGroups[102X( [3Xsize[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumberPerfectLibraryGroups[102X( [3Xsize[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrPerfectLibraryGroups[102X( [3Xsize[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the  number  of non-isomorphic perfect groups of size [3Xsize[103X for each
  positive  integer [3Xsize[103X up to [22X2⋅10^6[122X. Additionally, for odd [3Xsize[103X an answer is
  returned  (odd  order  groups  are  solvable). For any other argument out of
  range it returns [9Xfail[109X. [3XNrPerfectGroups[103X is a synonym for [2XNumberPerfectGroups[102X.
  Moreover [3XNumberPerfectLibraryGroups[103X (and its synonym [3XNrPerfectLibraryGroups[103X)
  exist  for  historical  reasons,  and return 0 instead of fail for arguments
  outside the library scope.[133X
  
  [1X1.1-5 SizeNumbersPerfectGroups[101X
  
  [33X[1;0Y[29X[2XSizeNumbersPerfectGroups[102X( [3Xfactor1[103X, [3Xfactor2[103X, [3X...[103X ) [32X function[133X
  
  [33X[0;0Y[2XSizeNumbersPerfectGroups[102X returns a list of pairs, each entry consisting of a
  group  order and the number of those groups in the library of perfect groups
  that  contain  the  specified  factors  [3Xfactor1[103X,  [3Xfactor2[103X,  ...  among their
  composition factors.[133X
  
  [33X[0;0YEach  argument  must  either  be the name of a nonabelian simple group or an
  integer  which  stands  for  the  product of the sizes of one or more cyclic
  factors. (In fact, the function replaces all integers among the arguments by
  their product.)[133X
  
  [33X[0;0YThe following text strings are accepted as simple group names.[133X
  
  [30X    [33X[0;6Y[10XA[3Xn[103X[10X[110X  or  [10XA([3Xn[103X[10X)[110X for the alternating groups [22XA_[3Xn[103X[122X, [22X5 ≤ n ≤ 9[122X, for example [10XA5[110X
        or [10XA(6)[110X.[133X
  
  [30X    [33X[0;6Y[10XL[3Xn[103X[10X([3Xq[103X[10X)[110X  or [10XL([3Xn[103X[10X,[3Xq[103X[10X)[110X for PSL[22X(n,q)[122X, where [22Xn ∈ { 2, 3 }[122X and [22Xq[122X a prime power,
        ranging[133X
  
        [30X    [33X[0;12Yfor [22Xn = 2[122X from 4 to 125[133X
  
        [30X    [33X[0;12Yfor [22Xn = 3[122X from 2 to 5[133X
  
  [30X    [33X[0;6Y[10XU[3Xn[103X[10X([3Xq[103X[10X)[110X  or [10XU([3Xn[103X[10X,[3Xq[103X[10X)[110X for PSU[22X(n,q)[122X, where [22Xn ∈ { 3, 4 }[122X and [22Xq[122X a prime power,
        ranging[133X
  
        [30X    [33X[0;12Yfor [22Xn = 3[122X from 3 to 5[133X
  
        [30X    [33X[0;12Yfor [22Xn = 4[122X from 2 to 2[133X
  
  [30X    [33X[0;6Y[10XSp4(4)[110X or [10XS(4,4)[110X for the symplectic group Sp[22X(4,4)[122X,[133X
  
  [30X    [33X[0;6Y[10XSz(8)[110X for the Suzuki group Sz[22X(8)[122X,[133X
  
  [30X    [33X[0;6Y[10XM[3Xn[103X[10X[110X or [10XM([3Xn[103X[10X)[110X for the Mathieu groups [22XM_11[122X, [22XM_12[122X, and [22XM_22[122X, and[133X
  
  [30X    [33X[0;6Y[10XJ[3Xn[103X[10X[110X or [10XJ([3Xn[103X[10X)[110X for the Janko groups [22XJ_1[122X and [22XJ_2[122X.[133X
  
  [33X[0;0YNote  that,  for most of the groups, the preceding list offers two different
  names  in order to be consistent with the notation used in [HP89] as well as
  with   the   notation   used  in  the  [2XDisplayCompositionSeries[102X  ([14XReference:
  DisplayCompositionSeries[114X) command of [5XGAP[105X. However, as the names are compared
  as  text  strings,  you are restricted to the above choice. Even expressions
  like [10XL2(2^5)[110X are not accepted.[133X
  
  [33X[0;0YAs  the use of the term PSU[22X(n,q)[122X is not unique in the literature, we mention
  that  in  this library it denotes the factor group of SU[22X(n,q)[122X by its centre,
  where  SU[22X(n,q)[122X  is  the  group of all [22Xn × n[122X unitary matrices with entries in
  [22XGF(q^2)[122X and determinant 1.[133X
  
  [33X[0;0YThe  purpose  of the function is to provide a simple way to formulate a loop
  over all library groups which contain certain composition factors.[133X
  
  [33X[0;0YAs  this  function  runs  over the whole library, it requires the data files
  that  are not distributed with this package, whatever its arguments are, see
  [14X1.2[114X.[133X
  
  
  [1X1.1-6 [33X[0;0YDisplayInformationPerfectGroups[133X[101X
  
  [33X[1;0Y[29X[2XDisplayInformationPerfectGroups[102X( [3Xsize[103X[, [3Xn[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XDisplayInformationPerfectGroups[102X( [3Xsizenumberpair[103X ) [32X function[133X
  
  [33X[0;0Y[2XDisplayInformationPerfectGroups[102X  displays  some invariants of the [3Xn[103X-th group
  of order [3Xsize[103X from the perfect groups library.[133X
  
  [33X[0;0YIf  no  value  of [3Xn[103X has been specified, the invariants will be displayed for
  all groups of size [3Xsize[103X available in the library.[133X
  
  [33X[0;0YAlternatively,  also  a  list  of  length  two  may  be  entered as the only
  argument, with entries [3Xsize[103X and [3Xn[103X.[133X
  
  [33X[0;0YThe information provided for [22XG[122X includes the following items:[133X
  
  [30X    [33X[0;6Ya  headline  containing  the  size number [10X[ [3Xsize[103X[10X, [3Xn[103X[10X ][110X of [22XG[122X in the form
        [10X[3Xsize[103X[10X.[3Xn[103X[10X[110X  (the  suffix [10X.[3Xn[103X[10X[110X will be suppressed if, up to isomorphism, [22XG[122X is
        the only perfect group of order [3Xsize[103X),[133X
  
  [30X    [33X[0;6Ya  message if [22XG[122X is simple or quasisimple, i.e., if the factor group of
        [22XG[122X by its centre is simple,[133X
  
  [30X    [33X[0;6Ythe  [21Xdescription[121X  of  the  structure  of  [22XG[122X as it is given by Holt and
        Plesken in [HP89] (see below),[133X
  
  [30X    [33X[0;6Ythe size of the centre of [22XG[122X (suppressed, if [22XG[122X is simple),[133X
  
  [30X    [33X[0;6Ythe prime decomposition of the size of [22XG[122X,[133X
  
  [30X    [33X[0;6Yorbit  sizes  for  a faithful permutation representation of [22XG[122X which is
        provided by the library (see below),[133X
  
  [30X    [33X[0;6Ya  reference  to  each occurrence of [22XG[122X in the tables of section 5.3 of
        [HP89].  Each  of  these  references consists of a class number and an
        internal  number [22X(i,j)[122X under which [22XG[122X is listed in that class. For some
        groups,  there  is more than one reference because these groups belong
        to more than one of the classes in the book.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayInformationPerfectGroups( 30720, 3 );[127X[104X
    [4X[28X#I Perfect group 30720:  A5 ( 2^4 E N 2^1 E 2^4 ) A[128X[104X
    [4X[28X#I   size = 2^11*3*5  orbit size = 240[128X[104X
    [4X[28X#I   Holt-Plesken class 1 (9,3)[128X[104X
    [4X[25Xgap>[125X [27XDisplayInformationPerfectGroups( 30720, 6 );[127X[104X
    [4X[28X#I Perfect group 30720:  A5 ( 2^4 x 2^4 ) C N 2^1[128X[104X
    [4X[28X#I   centre = 2  size = 2^11*3*5  orbit size = 384[128X[104X
    [4X[28X#I   Holt-Plesken class 1 (9,6)[128X[104X
    [4X[25Xgap>[125X [27XDisplayInformationPerfectGroups( Factorial( 8 ) / 2 );[127X[104X
    [4X[28X#I Perfect group 20160.1:  A5 x L3(2) 2^1[128X[104X
    [4X[28X#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 5 + 16[128X[104X
    [4X[28X#I   Holt-Plesken class 31 (1,1) (occurs also in class 32)[128X[104X
    [4X[28X#I Perfect group 20160.2:  A5 2^1 x L3(2)[128X[104X
    [4X[28X#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 7 + 24[128X[104X
    [4X[28X#I   Holt-Plesken class 31 (1,2) (occurs also in class 32)[128X[104X
    [4X[28X#I Perfect group 20160.3:  ( A5 x L3(2) ) 2^1[128X[104X
    [4X[28X#I   centre = 2  size = 2^6*3^2*5*7  orbit size = 192[128X[104X
    [4X[28X#I   Holt-Plesken class 31 (1,3)[128X[104X
    [4X[28X#I Perfect group 20160.4:  simple group  A8[128X[104X
    [4X[28X#I   size = 2^6*3^2*5*7  orbit size = 8[128X[104X
    [4X[28X#I   Holt-Plesken class 26 (0,1)[128X[104X
    [4X[28X#I Perfect group 20160.5:  simple group  L3(4)[128X[104X
    [4X[28X#I   size = 2^6*3^2*5*7  orbit size = 21[128X[104X
    [4X[28X#I   Holt-Plesken class 27 (0,1)[128X[104X
  [4X[32X[104X
  
  
  [1X1.2 [33X[0;0YData Not Distributed with the Package[133X[101X
  
  [33X[0;0YFor  reasons  of  size,  the  data  for  two orders is [13Xnot[113X shipped with this
  package:  the  [22X1639[122X  groups  of  order  [22X1376256[122X and the [22X7344[122X groups of order
  [22X1966080[122X.  All  other  orders  are  covered  by  the  data  files in the [11Xdata[111X
  subdirectory of the package.[133X
  
  [33X[0;0YHow many groups there are of these two orders is recorded in the index file,
  which    is    always    present,   so   [2XNumberPerfectGroups[102X   ([14X1.1-4[114X)   and
  [2XSizesPerfectGroups[102X ([14X1.1-1[114X) work as usual. Any function that has to touch the
  groups  themselves,  however,  enters a break loop with a message naming the
  file that is missing and where to get it:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := PerfectGroup(1376256, 1);[127X[104X
    [4X[28XError,[128X[104X
    [4X[28X[128X[104X
    [4X[28XFor reasons of size, the perfect groups library for orders >10^6 is[128X[104X
    [4X[28Xnot distributed fully by default. To access the group requested, get[128X[104X
    [4X[28Xthe file perf27.grp from[128X[104X
    [4X[28Xhttps://github.com/hulpke/extraperfect[128X[104X
    [4X[28Xand put it in the `data` subdirectory of the `perfgrp` package. Then type[128X[104X
    [4X[28X[128X[104X
    [4X[28Xreturn;[128X[104X
    [4X[28X[128X[104X
    [4X[28Xto continue in this GAP session (which will read in the file).[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo    proceed,    download    the   file   named   in   the   message   from
  [7Xhttps://github.com/hulpke/extraperfect[107X, place it in the [11Xdata[111X subdirectory of
  the  [5XPerfGrp[105X  package,  and  then  type [10Xreturn;[110X. The interrupted computation
  continues,  reading  in  the file that has just been added. The two files in
  question are [11Xperf27.grp[111X for order [22X1376256[122X and [11Xperf33.grp[111X for order [22X1966080[122X.[133X
  
  [33X[0;0YBesides  [2XPerfectGroup[102X  ([14X1.1-2[114X)  and  [2XDisplayInformationPerfectGroups[102X ([14X1.1-6[114X)
  called    with    one    of    these   two   orders,   this   also   affects
  [2XSizeNumbersPerfectGroups[102X  ([14X1.1-5[114X), which iterates over the whole library and
  therefore needs [13Xboth[113X files, whatever composition factors are asked for.[133X
  
  
  [1X1.3 [33X[0;0YMore about the Perfect Groups Library[133X[101X
  
  [33X[0;0YFor  any  library  group  [22XG[122X,  the  library  files  do  not  only  provide  a
  presentation,  but,  in  addition, a list of one or more subgroups [22XS_1, ...,
  S_r[122X  of  [22XG[122X  such that there is a faithful permutation representation of [22XG[122X of
  degree  [22X∑_{i  =  1}^r [G:S_i][122X on the set [22X{ S_i g ∣ 1 ≤ i ≤ r, g ∈ G }[122X of the
  cosets  of  the  [22XS_i[122X. This allows one to construct the groups as permutation
  groups.  The  function [2XDisplayInformationPerfectGroups[102X ([14X1.1-6[114X) displays only
  the available degree. The message[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28Xorbit size = 8[128X[104X
  [4X[32X[104X
  
  [33X[0;0Yin  the above example means that the available permutation representation is
  transitive and of degree 8, whereas the message[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28Xorbit sizes = 5 + 16[128X[104X
  [4X[32X[104X
  
  [33X[0;0Ymeans  that  a  nontransitive  permutation representation is available which
  acts on two orbits of size 5 and 16 respectively.[133X
  
  [33X[0;0YThe   notation   used  in  the  [21Xdescription[121X  of  a  group  is  explained  in
  section 5.1.2 of [HP89]. We quote the respective page from there:[133X
  
  [33X[0;0YWithin  a  class  [22XQ#p[122X,  an  isomorphism type of groups will be denoted by an
  ordered  pair  of integers [22X(r,n)[122X, where [22Xr ≥ 0[122X and [22Xn > 0[122X. More precisely, the
  isomorphism types in [22XQ # p[122X of order [22Xp^r |Q|[122X will be denoted by [22X(r,1), (r,2),
  (r,3), ...[122X. Thus [22XQ[122X will always get the size number [22X(0,1)[122X.[133X
  
  [33X[0;0YIn addition to the symbol [22X(r,n)[122X, the groups in [22XQ#p[122X will also be given a more
  descriptive name. The purpose of this is to provide a very rough idea of the
  structure of the group. The names are derived in the following manner. First
  of all, the isomorphism classes of irreducible [22XF_pQ[122X-modules [22XM[122X with [22X|Q|.|M| ≤
  10^6[122X,  where  [22XF_p[122X  is the field of order [22Xp[122X, are assigned symbols. These will
  either  be  simply [22Xp^x[122X, where [22Xx[122X is the dimension of the module, or, if there
  is  more  than  one isomorphism class of irreducible modules having the same
  dimension,  they  will  be  denoted  by[22Xp^x[122X, [22Xp^{x'}[122X, etc. The one-dimensional
  module with trivial [22XQ[122X-action will therefore be denoted by [22Xp^1[122X. These symbols
  will  be  listed  under  the  description  of  [22XQ[122X.  The  group  name consists
  essentially of a list of the composition factors working from the top of the
  group  downwards;  hence  it  always starts with the name of [22XQ[122X itself. (This
  convention  is  the most convenient in our context, but it is different from
  that  adopted  in the ATLAS [CCN+85], for example, where composition factors
  are  listed  in the reverse order. For example, we denote a group isomorphic
  to [22XSL(2,5)[122X by [22XA_5 2^1[122X rather than [22X2.A_5[122X.)[133X
  
  [33X[0;0YSome  other  symbols are used in the name, in order to give some idea of the
  relationship between these composition factors, and splitting properties. We
  shall now list these additional symbols.[133X
  
  [8X[22X×[122X[108X
        [33X[0;6Ybetween  two  factors  denotes  a  direct  product  of [22XF_pQ[122X-modules or
        groups.[133X
  
  [8XC[108X
        [33X[0;6Y(for [21Xcommutator[121X) between two factors means that the second lies in the
        commutator  subgroup  of  the  first. Similarly, a segment of the form
        [22X(f_1  ×  f_2)  C  f_3[122X  would mean that the factors [22Xf_1[122X and [22Xf_2[122X commute
        modulo [22Xf_3[122X and [22Xf_3[122X lies in [22X[f_1,f_2][122X.[133X
  
  [8XA[108X
        [33X[0;6Y(for  [21Xabelian[121X) between two factors indicates that the second is in the
        [22Xp[122Xth  power  (but not the commutator subgroup) of the first. [21XA[121X may also
        follow the factors, if bracketed.[133X
  
  [8XE[108X
        [33X[0;6Y(for  [21Xelementary  abelian[121X) between two factors indicates that together
        they generate an elementary abelian group (modulo subsequent factors),
        but that the resulting [22XF_p Q[122X-module extension does not split.[133X
  
  [8XN[108X
        [33X[0;6Y(for  [21Xnonsplit[121X)  before  a  factor  indicates  that [22XQ[122X (or possibly its
        covering  group)  splits  down  as far at this factor but not over the
        factor  itself.  So [21X[22XQ f_1 N f_2[122X[121X means that the normal subgroup [22Xf_1 f_2[122X
        of  the  group  has  no  complement  but, modulo [22Xf_2[122X, [22Xf_1[122X, does have a
        complement.[133X
  
  [33X[0;0YBrackets have their obvious meaning. Summarizing, we have:[133X
  
  [8X[22X×[122X[108X
        [33X[0;6Y= direct product;[133X
  
  [8XC[108X
        [33X[0;6Y= commutator subgroup;[133X
  
  [8XA[108X
        [33X[0;6Y= abelian;[133X
  
  [8XE[108X
        [33X[0;6Y= elementary abelian; and[133X
  
  [8XN[108X
        [33X[0;6Y= nonsplit.[133X
  
  [33X[0;0YHere are some examples.[133X
  
  [8X(i)[108X
        [33X[0;6Y[22XA_5  (2^4  E 2^1 E 2^4) A[122X means that the pairs [22X2^4 E 2^1[122X and [22X2^1 E 2^4[122X
        are both elementary abelian of exponent 4.[133X
  
  [8X(ii)[108X
        [33X[0;6Y[22XA_5  (2^4  E  2^1  A)  C  2^1[122X  means that [22XO_2(G)[122X is of symplectic type
        [22X2^{1+5}[122X, with Frattini factor group of type [22X2^4 E 2^1[122X. The [21XA[121X after the
        [22X2^1[122X  indicates that [22XG[122X has a central cyclic subgroup [22X2^1 A 2^1[122X of order
        4.[133X
  
  [8X(iii)[108X
        [33X[0;6Y[22XL_3(2)  ((2^1 E) × ( N 2^3 E 2^{3'} A) C) 2^{3'}[122X means that the [22X2^{3'}[122X
        factor at the bottom lies in the commutator subgroup of the pair [22X2^3 E
        2^{3'}[122X in the middle, but the lower pair [22X2^{3'} A 2^{3'}[122X is abelian of
        exponent  4.  There is also a submodule [22X2^1 E 2^{3'}[122X, and the covering
        group  [22XL_3(2) 2^1[122X of [22XL_3(2)[122X does not split over the [22X2^3[122X factor. (Since
        [22XG[122X  is  perfect,  it  goes without saying that the extension [22XL_3(2) 2^1[122X
        cannot split itself.)[133X
  
  [33X[0;0YWe  must  stress that this notation does not always succeed in being precise
  or even unambiguous, and the reader is free to ignore it if it does not seem
  helpful.[133X
  
  [33X[0;0YIf  such a group description has been given in the book for [22XG[122X (and, in fact,
  this  is  the  case  for  most  of  the  library groups), it is displayed by
  [2XDisplayInformationPerfectGroups[102X  ([14X1.1-6[114X).  Otherwise the function provides a
  less  explicit description of the (in these cases unique) Holt-Plesken class
  to  which  [22XG[122X  belongs, together with a serial number if this is necessary to
  make it unique.[133X
  
