  
  
                                    [1X Cryst [101X
  
  
                    [1X Computing with crystallographic groups [101X
  
  
                                     4.1.32
  
  
                                 18 August 2026
  
  
                                  Bettina Eick
  
                                  Franz Gähler
  
                                 Werner Nickel
  
  
  
  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, TU Braunschweig[133X
                [33X[0;14YUniversitätsplatz 2, D-38106 Braunschweig, Germany[133X
  
  
  Franz Gähler
      Email:    [7Xmailto:gaehler@math.uni-bielefeld.de[107X
      Homepage: [7Xhttps://www.math.uni-bielefeld.de/~gaehler/[107X
      Address:  [33X[0;14YFakultät für Mathematik, Universität Bielefeld[133X
                [33X[0;14YPostfach 10 01 31, D-33501 Bielefeld, Germany[133X
  
  
  Werner Nickel
      Email:    [7Xmailto:nickel@mathematik.tu-darmstadt.de[107X
  
  -------------------------------------------------------
  
  
  [1XContents (Cryst)[101X
  
  1 [33X[0;0YIntroduction[133X
  2 [33X[0;0YAffine crystallographic groups[133X
    2.1 [33X[0;0YThe default action[133X
      2.1-1 SetCrystGroupDefaultAction
    2.2 [33X[0;0YConstruction[133X
      2.2-1 AffineCrystGroupOnRight
      2.2-2 AsAffineCrystGroupOnRight
      2.2-3 IsAffineCrystGroupOnRight
      2.2-4 AffineCrystGroupOnLeft
      2.2-5 AsAffineCrystGroupOnLeft
      2.2-6 IsAffineCrystGroupOnLeft
      2.2-7 AffineCrystGroup
      2.2-8 AsAffineCrystGroup
      2.2-9 IsAffineCrystGroup
      2.2-10 TransposedMatrixGroup
    2.3 [33X[0;0YPoint group[133X
      2.3-1 PointGroup
      2.3-2 IsPointGroup
      2.3-3 AffineCrystGroupOfPointGroup
      2.3-4 PointHomomorphism
      2.3-5 IsPointHomomorphism
    2.4 [33X[0;0YTranslation lattice[133X
      2.4-1 TranslationBasis
      2.4-2 InternalBasis
      2.4-3 AddTranslationBasis
      2.4-4 CheckTranslationBasis
      2.4-5 IsSpaceGroup
      2.4-6 IsStandardAffineCrystGroup
      2.4-7 IsStandardSpaceGroup
      2.4-8 StandardAffineCrystGroup
      2.4-9 IsSymmorphicSpaceGroup
    2.5 [33X[0;0YSpecial methods[133X
      2.5-1 \^
      2.5-2 IsomorphismFpGroup
      2.5-3 IsomorphismFpGroup
      2.5-4 IsomorphismPcpGroup
      2.5-5 IsomorphismPcpGroup
    2.6 [33X[0;0YMaximal subgroups[133X
      2.6-1 MaximalSubgroupClassReps
      2.6-2 ConjugacyClassesMaximalSubgroups
    2.7 [33X[0;0YSpace groups with a given point group[133X
      2.7-1 SpaceGroupsByPointGroupOnRight
      2.7-2 SpaceGroupTypesByPointGroupOnRight
      2.7-3 SpaceGroupsByPointGroupOnLeft
      2.7-4 SpaceGroupTypesByPointGroupOnLeft
      2.7-5 SpaceGroupsByPointGroup
      2.7-6 SpaceGroupTypesByPointGroup
    2.8 [33X[0;0YWyckoff positions[133X
      2.8-1 WyckoffPositions
      2.8-2 WyckoffPositionsByStabilizer
      2.8-3 IsWyckoffPosition
      2.8-4 WyckoffBasis
      2.8-5 WyckoffTranslation
      2.8-6 WyckoffSpaceGroup
      2.8-7 WyckoffStabilizer
      2.8-8 WyckoffOrbit
      2.8-9 WyckoffGraph
    2.9 [33X[0;0YNormalizers[133X
      2.9-1 NormalizerPointGroupInGLnZ
      2.9-2 CentralizerPointGroupInGLnZ
      2.9-3 TranslationNormalizer
      2.9-4 AffineNormalizer
      2.9-5 AffineInequivalentSubgroups
      2.9-6 ConjugatorSpaceGroups
    2.10 [33X[0;0YColor groups[133X
      2.10-1 ColorGroup
      2.10-2 IsColorGroup
      2.10-3 ColorSubgroup
      2.10-4 ColorCosetList
      2.10-5 ColorOfElement
      2.10-6 ColorPermGroup
      2.10-7 ColorHomomorphism
      2.10-8 Subgroup
    2.11 [33X[0;0YColored AffineCrystGroups[133X
      2.11-1 PointGroup
    2.12 [33X[0;0YInternational Tables[133X
      2.12-1 SpaceGroupSettingsIT
      2.12-2 SpaceGroupOnRightIT
      2.12-3 SpaceGroupOnLeftIT
      2.12-4 SpaceGroupIT
  
  
  [32X
