  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  
  [1X1.1 [33X[0;0YOverview[133X[101X
  
  [33X[0;0YThis manual describes the [5XCubefree[105X package, a [5XGAP[105X 4 package for constructing
  groups  of  cubefree order; i.e., groups whose order is not divisible by any
  third power of a prime.[133X
  
  [33X[0;0YThe  groups  of  squarefree  order  have been known for a long time: Hoelder
  [Hoe95]  investigated them at the end of the 19th century. Taunt [Tau55] has
  considered  solvable  groups  of  cubefree order, since he examined solvable
  groups  with  abelian  Sylow  subgroups.  Cubefree  groups  in  general  are
  investigated  firstly  in  [Die05],  [DE05],  and  [DE12],  and this package
  contains the implementation of the algorithms described there.[133X
  
  [33X[0;0YSome  general  approaches  to construct groups of an arbitrarily given order
  are described in [BE99a], [BE99b], and [BEO02].[133X
  
  [33X[0;0YThe  main  function of this package is a method to construct all groups of a
  given  cubefree  order up to isomorphism. The algorithm behind this function
  is  described  completely  in  [Die05] and [DE05]. It is a refinement of the
  methods of the [5XGrpConst[105X package which are described in [BE99c].[133X
  
  [33X[0;0YThis  main function needs a method to construct up to conjugacy the solvable
  cubefree  subgroups of GL[22X(2,p)[122X coprime to [22Xp[122X. We split this construction into
  the  construction  of  reducible  and  irreducible  subgroups of GL[22X(2,p)[122X. To
  determine  the  irreducible  subgroups we use the method described in [FO05]
  for  which  this package also contains an implementation. Alternatively, the
  [5XIrredSol[105X package [Hoe00] could be used for primes [22Xp≤ 251[122X.[133X
  
  [33X[0;0YThe   algorithm   of   [FO05]   requires   a  method  to  rewrite  a  matrix
  representation. We use and implement the method of [GH97] for this purpose.[133X
  
  [33X[0;0YOne  can  modify  the  construction  algorithm for cubefree groups to a very
  efficient algorithm to construct groups of squarefree order. This is already
  done  in  the  [5XSmallGroups[105X  library.  Thus for the construction of groups of
  squarefree  order  it  is  more  practical  to  use  [10XAllSmallGroups[110X  of  the
  [5XSmallGroups[105X library.[133X
  
  [33X[0;0YA  more  detailed  description  of  the  implemented methods can be found in
  Chapter [14X2[114X.[133X
  
  [33X[0;0YChapter [14X3[114X explains how to install and load the [5XCubefree[105X package.[133X
  
  
  [1X1.2 [33X[0;0YTheoretical background[133X[101X
  
  [33X[0;0YIn  this  section  we  give a brief survey about the main algorithm which is
  used  to  construct groups of cubefree order: the Frattini extension method.
  For  a  by  far  more detailed description we refer to the above references;
  e.g. see the online version of [Die05].[133X
  
  [33X[0;0YLet  [22XG[122X  be  a  finite group. The Frattini subgroup [22XΦ(G)[122X is defined to be the
  intersection  of  all maximal subgroups of [22XG[122X. We say a group [22XH[122X is a Frattini
  extension  by  [22XG[122X  if  the  Frattini  factor  [22XH/Φ(H)[122X  is isomorphic to [22XG[122X. The
  Frattini  factor  of  [22XH[122X  is  Frattini-free;  i.e.  it has a trivial Frattini
  subgroup.  It  is known that every prime divisor of [22X|H|[122X is also a divisor of
  [22X|H/Φ(H)|[122X.  Thus  the  Frattini  subgroup  of  a  cubefree  group  has  to be
  squarefree  and, as it is nilpotent, it is a direct product of cyclic groups
  of prime order.[133X
  
  [33X[0;0YHence in order to construct all groups of a given cubefree order [22Xn[122X, say, one
  can,  firstly,  construct  all  Frattini-free groups of suitable orders and,
  secondly,  compute all corresponding Frattini extensions of order [22Xn[122X. A first
  fundamental  result  is  that a group of cubefree order is either a solvable
  Frattini  extension  or  a direct product of a PSL[22X(2,r)[122X, [22Xr>3[122X a prime, with a
  solvable  Frattini  extension.  In particular, the simple groups of cubefree
  order  are  the groups PSL[22X(2,r)[122X with [22Xr>3[122X a prime such that [22Xr± 1[122X is cubefree.
  As  a  nilpotent  group  is the direct product of its Sylow subgroups, it is
  straightforward to compute all nilpotent groups of a given cubefree order.[133X
  
  [33X[0;0YAnother important result is that for a cubefree solvable Frattini-free group
  there is exactly one isomorphism type of suitable Frattini extensions, which
  restricts  the  construction  of  cubefree  groups  to  the determination of
  cubefree   solvable   Frattini-free  groups.  This  uniqueness  of  Frattini
  extensions  is  the  main  reason why the Frattini extension method works so
  efficiently in the cubefree case.[133X
  
  [33X[0;0YIn  other  words,  there is a one-to-one correspondence between the solvable
  cubefree  groups  of order [22Xn[122X and some Frattini-free groups of order dividing
  [22Xn[122X.  This  allows  one  to  count the number of isomorphism types of cubefree
  groups of a given order without constructing Frattini extensions.[133X
  
  [33X[0;0YIn  the  remaining  part of this section we consider the construction of the
  solvable  Frattini-free  groups of a given cubefree order up to isomorphism.
  Such  a  group is a split extension over its socle; i.e. over the product of
  its  minimal  normal  subgroups.  Let [22XF[122X be a solvable Frattini-free group of
  cubefree order with socle [22XS[122X. Then [22XS[122X is a (cubefree) direct product of cyclic
  groups  of  prime  order  and  [22XF[122X  can be written as [22XF=K⋉ S[122X where [22XK≤[122XAut[22X(S)[122X is
  determined  up  to  conjugacy.  In  particular,  [22XK[122X is a subdirect product of
  certain  cubefree subgroups of groups of the type GL[22X(2,p)[122X or [22XC_p-1[122X. Hence in
  order  to  determine all possible subgroups [22XK[122X one can determine all possible
  projections  from  such  a  subgroup  into  the  direct factors of the types
  GL[22X(2,p)[122X  and  [22XC_p-1[122X,  and  then  form  all  subdirect  products having these
  projections. The construction of these subdirect products is one of the most
  time-consuming parts in the Frattini extension method for cubefree groups.[133X
  
