  
  [1X2 [33X[0;0YThe automorphism group method[133X[101X
  
  
  [1X2.1 [33X[0;0YThe automorphism group method[133X[101X
  
  [33X[0;0YThe  [5XAutPGrp[105X  package  installs  a method for [10XAutomorphismGroup[110X for a finite
  [22Xp[122X-group  (see  also  Section [14X'Reference: Groups of Automorphisms'[114X in the [5XGAP[105X
  Reference Manual).[133X
  
  [1X2.1-1 AutomorphismGroup[101X
  
  [33X[1;0Y[29X[2XAutomorphismGroup[102X( [3XG[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe automorphism group of [3XG[103X.[133X
  
  [33X[0;0YThe  input  is  a  finite  [22Xp[122X-group  [3XG[103X. If the filters [10XIsPGroup[110X, [10XIsFinite[110X and
  [10XCanEasilyComputePcgs[110X  are  set and true for [3XG[103X, the method selection of [5XGAP[105X 4
  invokes this algorithm.[133X
  
  [33X[0;0YThe  output  of  the  method  is an automorphism group, whose generators are
  given  in  [10XGroupHomomorphismByImages[110X  format in terms of their action on the
  underlying group [3XG[103X.[133X
  
  [1X2.1-2 InfoAutGrp[101X
  
  [33X[1;0Y[29X[2XInfoAutGrp[102X [32X info class[133X
  
  [33X[0;0YThis  is  a  [5XGAP[105X  [2XInfoClass[102X  ([14XReference:  InfoClass  for  a GAP package[114X). By
  assigning an [10Xlevel[110X in the range 1 to 4 via[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XSetInfoLevel(InfoAutGrp, level);[128X[104X
  [4X[32X[104X
  
  [33X[0;0Yvarying  levels  of  information on the progress of the computation, will be
  obtained.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage("autpgrp", false);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XG := PcGroupCode(619031068735, 32);  # SmallGroup( 32, 15 );[127X[104X
    [4X[28X<pc group of size 32 with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XSetInfoLevel( InfoAutGrp, 1 );[127X[104X
    [4X[25Xgap>[125X [27XAutomorphismGroup(G);[127X[104X
    [4X[28X#I  step 1: 2^2 -- init automorphisms[128X[104X
    [4X[28X#I  step 2: 2^2 -- aut grp has size 2[128X[104X
    [4X[28X#I  step 3: 2^1 -- aut grp has size 32[128X[104X
    [4X[28X#I  final step: convert[128X[104X
    [4X[28X<group of size 64 with 6 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe algorithm proceeds by induction down the lower [22Xp[122X-central series of [10XG[110X and
  the information corresponds to the steps of this induction. In the following
  example we observe that the method also accepts permutation groups as input,
  provided they satisfy the required filters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := DihedralGroup( IsPermGroup, 2^5 );;[127X[104X
    [4X[25Xgap>[125X [27XIsPGroup(G);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XCanEasilyComputePcgs(G);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsFinite(G);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XA := AutomorphismGroup(G);[127X[104X
    [4X[28X#I  step 1: 2^2 -- init automorphisms[128X[104X
    [4X[28X#I  step 2: 2^1 -- aut grp has size 2[128X[104X
    [4X[28X#I  step 3: 2^1 -- aut grp has size 8[128X[104X
    [4X[28X#I  step 4: 2^1 -- aut grp has size 32[128X[104X
    [4X[28X#I  final step: convert[128X[104X
    [4X[28X<group of size 128 with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27XA.1;[127X[104X
    [4X[28XPcgs([ ( 2,16)( 3,15)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10),[128X[104X
    [4X[28X  ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),[128X[104X
    [4X[28X  ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16),[128X[104X
    [4X[28X  ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16),[128X[104X
    [4X[28X  ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]) ->[128X[104X
    [4X[28X[ ( 1, 2)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10),[128X[104X
    [4X[28X  ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),[128X[104X
    [4X[28X  ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16),[128X[104X
    [4X[28X  ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16),[128X[104X
    [4X[28X  ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ][128X[104X
    [4X[25Xgap>[125X [27XOrder(A.1);[127X[104X
    [4X[28X16[128X[104X
  [4X[32X[104X
  
