  
  
                                   [1X AutPGrp [101X
  
  
                [1X Computing the Automorphism Group of a p-Group [101X
  
  
                                     1.12.0
  
  
                                  17 May 2026
  
  
                                  Bettina Eick
  
                                 Eamonn O'Brien
  
  
  
  Bettina Eick
      Email:    [7Xmailto:beick@tu-bs.de[107X
      Homepage: [7Xhttp://www.iaa.tu-bs.de/beick[107X
      Address:  [33X[0;14YInstitut Analysis und Algebra[133X
                [33X[0;14YTU Braunschweig[133X
                [33X[0;14YUniversitätsplatz 2[133X
                [33X[0;14YD-38106 Braunschweig[133X
                [33X[0;14YGermany[133X
  
  
  Eamonn O'Brien
      Email:    [7Xmailto:obrien@math.auckland.ac.nz[107X
      Homepage: [7Xhttp://www.math.auckland.ac.nz/~obrien[107X
      Address:  [33X[0;14YDepartment of Mathematics[133X
                [33X[0;14YUniversity of Auckland[133X
                [33X[0;14YPrivate Bag 92019[133X
                [33X[0;14YAuckland[133X
                [33X[0;14YNew Zealand[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  [5XAutPGrp[105X  package  introduces a new function to compute the automorphism
  group  of  a finite [22Xp[122X-group. The underlying algorithm is a refinement of the
  methods  described  in O'Brien (1995). In particular, this implementation is
  more  efficient  in  both  time and space requirements and hence has a wider
  range  of  applications than the ANUPQ method. Our package is written in [5XGAP[105X
  code  and  it  makes use of a number of methods from the [5XGAP[105X library such as
  the  MeatAxe  for  matrix  groups  and  permutation group functions. We have
  compared  our  method  to  the  others available in [5XGAP[105X. Our package usually
  out-performs  all but the method designed for finite abelian groups. We note
  that our method uses the small groups library in certain cases and hence our
  algorithm is more effective if the small groups library is installed.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0YBettina Eick and Eamonn O'Brien.[133X
  
  [33X[0;0YAutPGrp is free software; you can redistribute it and/or modify it under the
  terms        of       the       GNU       General       Public       License
  ([7Xhttps://www.fsf.org/licenses/gpl.html[107X)  as  published  by the Free Software
  Foundation;  either  version 2 of the License, or (at your option) any later
  version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe  thank  Alexander  Hulpke for helping us with efficiency problems. Werner
  Nickel provided some functions from the [5XGAP[105X [10XPQuotient[110X which are used in this
  package.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (AutPGrp)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YIntroduction[133X
  2 [33X[0;0YThe automorphism group method[133X
    2.1 [33X[0;0YThe automorphism group method[133X
      2.1-1 AutomorphismGroup
      2.1-2 InfoAutGrp
  3 [33X[0;0YThe underlying function[133X
    3.1 [33X[0;0YThe underlying function[133X
      3.1-1 AutomorphismGroupPGroup
      3.1-2 ConvertHybridAutGroup
      3.1-3 PcGroupAutPGroup
  4 [33X[0;0YInfluencing the algorithm[133X
    4.1 [33X[0;0YOutline of the algorithm[133X
    4.2 [33X[0;0YThe initialisation step[133X
    4.3 [33X[0;0YStabilisers in matrix groups[133X
      4.3-1 CHOP_MULT
    4.4 [33X[0;0YSearching for a small generating set[133X
      4.4-1 NICE_STAB
    4.5 [33X[0;0YAn interactive version of the algorithm[133X
  5 [33X[0;0YAdditional features of the package[133X
    5.1 [33X[0;0YAdditional features of the package[133X
      5.1-1 NumberOfPClass2PGroups
      5.1-2 NumberOfPClass2PGroups
      5.1-3 NumberOfClass2LieAlgebras
      5.1-4 NumberOfClass2LieAlgbras
  
  
  [32X
