  
  [1X2 [33X[0;0YFunctionality of the Cubefree package[133X[101X
  
  [33X[0;0YThis chapter describes the methods available from the [5XCubefree[105X package.[133X
  
  
  [1X2.1 [33X[0;0YNew methods[133X[101X
  
  [33X[0;0YThis section lists the implemented functions.[133X
  
  [1X2.1-1 ConstructAllCFGroups[101X
  
  [33X[1;0Y[29X[2XConstructAllCFGroups[102X( [3Xorder[103X ) [32X function[133X
  
  [33X[0;0YThe  input  [3Xorder[103X  has  to  be  a positive cubefree integer. The output is a
  complete  and irredundant list of isomorphism type representatives of groups
  of  this  size.  If  possible,  the  groups  are  given  as pc groups and as
  permutation groups otherwise.[133X
  
  [1X2.1-2 ConstructAllCFSolvableGroups[101X
  
  [33X[1;0Y[29X[2XConstructAllCFSolvableGroups[102X( [3Xorder[103X ) [32X function[133X
  
  [33X[0;0YThe  input  [3Xorder[103X  has  to  be  a positive cubefree integer. The output is a
  complete  and  irredundant  list  of  isomorphism  type  representatives  of
  solvable groups of this size. The groups are given as pc groups.[133X
  
  [1X2.1-3 ConstructAllCFNilpotentGroups[101X
  
  [33X[1;0Y[29X[2XConstructAllCFNilpotentGroups[102X( [3Xorder[103X ) [32X function[133X
  
  [33X[0;0YThe  input  [3Xorder[103X  has  to  be  a positive cubefree integer. The output is a
  complete  and  irredundant  list  of  isomorphism  type  representatives  of
  nilpotent groups of this size. The groups are given as pc groups.[133X
  
  [1X2.1-4 ConstructAllCFSimpleGroups[101X
  
  [33X[1;0Y[29X[2XConstructAllCFSimpleGroups[102X( [3Xorder[103X ) [32X function[133X
  
  [33X[0;0YThe  input  [3Xorder[103X  has  to  be  a positive cubefree integer. The output is a
  complete  and irredundant list of isomorphism type representatives of simple
  groups  of this size. In particular, there exists either none or exactly one
  simple group of the given order.[133X
  
  [1X2.1-5 ConstructAllCFFrattiniFreeGroups[101X
  
  [33X[1;0Y[29X[2XConstructAllCFFrattiniFreeGroups[102X( [3Xorder[103X ) [32X function[133X
  
  [33X[0;0YThe  input  [3Xorder[103X  has  to  be  a positive cubefree integer. The output is a
  complete  and  irredundant  list  of  isomorphism  type  representatives  of
  Frattini-free groups of this size.[133X
  
  [1X2.1-6 IsomorphismCubefreeGroups[101X
  
  [33X[1;0Y[29X[2XIsomorphismCubefreeGroups[102X( [3XG[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0YReturns  an  isomorphism between two cubefree groups [3XG[103X and [3XH[103X, if exists, and
  fail  otherwise.  It is assumed that the input groups are permutation groups
  or  pc  groups. The algorithm is currently efficient only for solvable input
  groups  due  to  the  lack  of  a constructive recognition algorithm for the
  simple factors PSL.[133X
  
  [1X2.1-7 IsIsomorphicCubefreeGroups[101X
  
  [33X[1;0Y[29X[2XIsIsomorphicCubefreeGroups[102X( [3XG[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0YReturns  true/false,  depending  on  whether two cubefree groups [3XG[103X and [3XH[103X are
  isomorphic. It is assumed that the input groups are permutation groups or pc
  groups.[133X
  
  [1X2.1-8 NumberCFGroups[101X
  
  [33X[1;0Y[29X[2XNumberCFGroups[102X( [3Xn[103X[, [3Xbool[103X] ) [32X function[133X
  
  [33X[0;0YThe  input  [3Xn[103X  has  to  be a positive cubefree integer and the output is the
  number  of  all  cubefree groups of order [3Xn[103X. The [5XSmallGroups[105X library is used
  for  squarefree orders, orders of the type [22Xp^2[122X and [22Xp^2q[122X, and cubefree orders
  less  than  50000.  Only  if  [3Xbool[103X is set to false, then only the squarefree
  orders  and  orders of the type [22Xp^2[122X and [22Xp^2q[122X, are taken from the [5XSmallGroups[105X
  library.[133X
  
  [1X2.1-9 NumberCFSolvableGroups[101X
  
  [33X[1;0Y[29X[2XNumberCFSolvableGroups[102X( [3Xn[103X[, [3Xbool[103X] ) [32X function[133X
  
  [33X[0;0YThe  input  [3Xn[103X  has  to  be a positive cubefree integer and the output is the
  number  of  all cubefree solvable groups of order [3Xn[103X. The [5XSmallGroups[105X library
  is used for squarefree orders and orders of the type [22Xp^2[122X and [22Xp^2q[122X, where all
  groups  are solvable. For other orders counting is faster than searching the
  library, so [3Xbool[103X has no effect there; it is accepted for compatibility.[133X
  
  [1X2.1-10 CountAllCFGroupsUpTo[101X
  
  [33X[1;0Y[29X[2XCountAllCFGroupsUpTo[102X( [3Xn[103X[, [3Xbool[103X] ) [32X function[133X
  
  [33X[0;0YThe  input is a positive integer [3Xn[103X and the output is a list [22XL[122X of size [3Xn[103X such
  that [22XL[i][122X contains the number of isomorphism types of groups of order [22Xi[122X if [22Xi[122X
  is  cubefree  and  [22XL[i][122X  is  not bound, otherwise, [22X1≤ i ≤ n[122X. The [5XSmallGroups[105X
  library  is used for squarefree orders, orders of the type [22Xp^2[122X and [22Xp^2q[122X, and
  cubefree orders less than 50000. Only if [3Xbool[103X is set to false, then only the
  squarefree  orders  and  orders  of the type [22Xp^2[122X and [22Xp^2q[122X are taken from the
  [5XSmallGroups[105X  library. This is a plain loop over [10XNumberCFGroups[110X, which caches
  the data it computes and hence shares most of the work between the orders.[133X
  
  [1X2.1-11 CubefreeOrderInfo[101X
  
  [33X[1;0Y[29X[2XCubefreeOrderInfo[102X( [3Xn[103X[, [3Xbool[103X] ) [32X function[133X
  
  [33X[0;0YThis function displays some (very vague) information about the complexity of
  the  construction of the groups of (cubefree) order [3Xn[103X. It returns the number
  of possible pairs [3X(a,b)[103X where [3Xa[103X is the order of a Frattini-free group [3XF[103X with
  socle  [3XS[103X  of  order  [3Xb[103X which has to be constructed in order to construct all
  groups  of order [3Xn[103X: In fact, for each of these pairs [3X(a,b)[103X one would have to
  construct  up  to conjugacy all subgroups of order [3Xa[103X/[3Xb[103X of Aut[3X(S)[103X. The sum of
  the numbers of these subgroups for all pairs [3X(a,b)[103X as above is the number of
  groups  of  order [3Xn[103X. Thus the output of [10XCubefreeOrderInfo[110X is a trivial lower
  bound  for  the  number  of  groups  of  order  [3Xn[103X.  There  is  no additional
  information displayed if [3Xbool[103X is set to false.[133X
  
  [1X2.1-12 CubefreeTestOrder[101X
  
  [33X[1;0Y[29X[2XCubefreeTestOrder[102X( [3Xn[103X ) [32X function[133X
  
  [33X[0;0YThe  input  has  to be a cubefree integer between 1 and 50000. This function
  tests  the  functionality of [5XCubefree[105X, i.e. functions (1)--(7), and compares
  it  with  the data of the [5XSmallGroups[105X library. It returns true if everything
  is okay, otherwise an error message will be displayed.[133X
  
  [1X2.1-13 IsCubeFreeInt[101X
  
  [33X[1;0Y[29X[2XIsCubeFreeInt[102X( [3Xn[103X ) [32X property[133X
  
  [33X[0;0YThe output is [3Xtrue[103X if [3Xn[103X is a cubefree integer and [3Xfalse[103X otherwise.[133X
  
  [1X2.1-14 IsSquareFreeInt[101X
  
  [33X[1;0Y[29X[2XIsSquareFreeInt[102X( [3Xn[103X ) [32X property[133X
  
  [33X[0;0YThe output is [3Xtrue[103X if [3Xn[103X is a squarefree integer and [3Xfalse[103X otherwise.[133X
  
  [1X2.1-15 IrreducibleSubgroupsOfGL[101X
  
  [33X[1;0Y[29X[2XIrreducibleSubgroupsOfGL[102X( [3Xn[103X, [3Xq[103X ) [32X operation[133X
  
  [33X[0;0YThe  current version of this function allows only [3Xn[103X=2. The input [3Xq[103X has to be
  a  prime-power  [3Xq[103X[22X=p^r[122X  with  [22Xp≥  5[122X  a  prime.  The  output  is a list of all
  irreducible subgroups of GL[22X(2,q)[122X up to conjugacy.[133X
  
  [1X2.1-16 RewriteAbsolutelyIrreducibleMatrixGroup[101X
  
  [33X[1;0Y[29X[2XRewriteAbsolutelyIrreducibleMatrixGroup[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0YThe  input  [22XG[122X has to be an absolutely irreducible matrix group over a finite
  field  GF[22X(q)[122X.  If  possible,  the output is [22XG[122X rewritten over the subfield of
  GF[22X(q)[122X  generated  by  the  traces  of  the elements of [22XG[122X. If no rewriting is
  possible, then the input [22XG[122X is returned.[133X
  
  
  [1X2.2 [33X[0;0YComments on the implementation[133X[101X
  
  [33X[0;0YThis section provides some information about the implementations.[133X
  
  [33X[0;0Y[13XConstructAllCFGroups[113X[133X
  
  [33X[0;0YThe  function [10XConstructAllCFGroups[110X constructs all groups of a given cubefree
  order  up to isomorphism using the Frattini Extension Method as described in
  [Die05],  [DE05],  [BE99a],  and [BE99b]. One step in the Frattini Extension
  Method  is  to compute Frattini extensions and for this purpose some already
  implemented methods of the required [5XGAP[105X  package [5XGrpConst[105X  are used.[133X
  
  [33X[0;0YSince  [10XConstructAllCFGroups[110X  requires only some special types of irreducible
  subgroups  of  GL[22X(2,p)[122X  (e.g.  of  cubefree  order),  it contains a modified
  internal  version of [10XIrreducibleSubgroupsOfGL[110X. This means that the latter is
  not called explicitly by [10XConstructAllCFGroups[110X.[133X
  
  [33X[0;0Y[13XConstructAllCFSimpleGroups and ConstructAllCFNilpotentGroups[113X[133X
  
  [33X[0;0YThe  construction  of simple or nilpotent groups of cubefree order is rather
  easy,  see [Die05] or [DE05]. In particular, the methods used in these cases
  are independent of the methods used in the general cubefree case.[133X
  
  [33X[0;0Y[13XCountAllCFGroupsUpTo[113X[133X
  
  [33X[0;0YAs described in [Die05] and [DE05], every cubefree group [22XG[122X has the form [22XG=A×
  I[122X where [22XA[122X is trivial or non-abelian simple and [22XI[122X is solvable. Further, there
  is a one-to-one correspondence between the solvable cubefree groups and [3Xsome[103X
  solvable  Frattini-free groups. This one-to-one correspondence allows one to
  count  the  number of groups of a given cubefree order without computing any
  Frattini   extension.  To  reduce  runtime,  the  computed  irreducible  and
  reducible subgroups of the general linear groups GL[22X(2,p)[122X and also the number
  of  the  computed  solvable Frattini-free groups are stored during the whole
  computation.  This  caching  is  done  by [10XNumberCFGroups[110X itself and persists
  between  calls,  so [10XCountAllCFGroupsUpTo[110X is just a loop over [10XNumberCFGroups[110X.
  Use  [10Xcf_ClearCountCaches()[110X  to  free the memory again. The solvable count is
  the  trivial-factor term of the total, so calling [10XNumberCFSolvableGroups[110X for
  an order for which [10XNumberCFGroups[110X has already run costs next to nothing.[133X
  
  [33X[0;0YCounting  never  constructs  a  group:  each socle complement determines one
  Frattini-free  group,  and  each  Frattini-free group one group of the given
  order. The subgroups of GL[22X(2,p)[122X are also replaced by small isomorphic copies
  carrying  the  action  of  their  normaliser,  so the subdirect products are
  computed  in  groups  of  order  a few thousand rather than in a permutation
  representation of GL[22X(2,p)[122X of degree [22Xp^2-1[122X.[133X
  
  [33X[0;0Y[13XIrreducibleSubgroupsOfGL[113X[133X
  
  [33X[0;0YIf  the  input  is  a  matrix  group over GF[22X(q)[122X, then the algorithm needs to
  construct GF[22X(q^3)[122X or GF[22X(q^6)[122X internally.[133X
  
  [33X[0;0Y[13XRewriteAbsolutelyIrreducibleMatrixGroup[113X[133X
  
  [33X[0;0YThe    function    [10XRewriteAbsolutelyIrreducibleMatrixGroup[110X    as   described
  algorithmically in [GH97] is a probabilistic Las Vegas algorithm; it retries
  until  a  correct  answer is returned. If the input is [22XG≤[122XGL[22X(d,p^r)[122X, then the
  expected runtime is [22XO(rd^3)[122X.[133X
  
  
  [1X2.3 [33X[0;0YComments on the efficiency[133X[101X
  
  [33X[0;0YThe  package  [5XGrpConst[105X  contains  several  implementations  of algorithms to
  construct  groups  of a given order. One of these algorithms is the Frattini
  extension  method,  see  Chapter [14X1[114X.  The  algorithm  used  in  [5XCubefree[105X is a
  modification  of  the  Frattini  extension  method  to  the case of cubefree
  orders.[133X
  
  [33X[0;0YThe  advantage  of  this modification is that the isomorphism problem at the
  construction  of  Frattini  extensions  is  solved completely on a theoretic
  level.  Also, the construction of the Frattini-free groups up to isomorphism
  is  reduced  to the determination of certain subgroups of groups of the type
  GL[22X(2,p)[122X  and [22XC_p-1[122X, [22Xp[122X a prime, and to the construction of subdirect products
  of these subgroups. As this is exponential, this is a main bottleneck of the
  current implementation.[133X
  
  [33X[0;0YA  modification of the Frattini extension method to squarefree orders yields
  a  powerful  construction  algorithm for squarefree groups which is based on
  number  theory only. An implementation of this algorithm can be found in the
  [5XSmallGroups[105X  library.  Thus  for squarefree groups one should definitely use
  [10XAllSmallGroups[110X  and  [10XNumberSmallGroups[110X instead of the functions of [5XCubefree[105X.
  The same holds for groups of order [22Xp^2[122X or [22Xp^2q[122X.[133X
  
  [33X[0;0YMoreover,  using  the functionality of [5XCubefree[105X, the [5XSmallGroups[105X library now
  contains  all  groups  of  cubefree order at most 50000. Hence, also in this
  case,  one  should prefer [10XAllSmallGroups[110X and [10XNumberSmallGroups[110X to access the
  data of the library directly.[133X
  
  [33X[0;0YFor  all other cubefree orders [3Xn[103X one can try to use [5XCubefree[105X to construct or
  count the corresponding groups. Note, that the success of these computations
  depends  basically  on  the  complexity and number theory of the prime-power
  factorization of [3Xn[103X. For each prime [22Xp[122X with [22Xp^2∣ n[122X one might have to construct
  subgroups  of  GL[22X(2,p)[122X and subdirect products involving these subgroups. One
  can   use  the  info  class  [10XInfoCF[110X  to  get  some  information  during  the
  computation. In order to construct subdirect products, we need a permutation
  representation of these matrix groups. To rewrite them at once, we compute a
  permutation  representation  of  GL[22X(2,p)[122X  and  apply this isomorphism to the
  constructed   subgroups.  Unfortunately,  this  is  quite  time  and  memory
  consuming for bigger primes.[133X
  
  [33X[0;0YIn other words, [5XCubefree[105X can not handle [13Xunreasonable[113X cubefree orders. To get
  a  rough  idea of the complexity of the computation of groups of order [22Xn[122X and
  to  get  a  trivial  lower  bound  for  the  number  of  groups, one can use
  [10XCubefreeOrderInfo(n)[110X.[133X
  
  [33X[0;0YAt   the   end   of   this   section   we  consider  the  quotient  [22Xq(n)[122X  of
  [10XNumberSmallGroups(n)[110X  and  [10XCubefreeOrderInfo(n)[110X  for  cubefree  [22X1≤ n≤ 50000[122X.
  Although for most of these integers we have a small quotient, note that [22Xq(n)[122X
  seems  to be unbounded in general. There are 41597 cubefree integers between
  [22X1[122X  and  [22X50000[122X and [22X26414[122X of these integers fulfill [22Xq(n)=1[122X. Moreover, 13065 of
  these  integers  fulfill  [22X1< q(n)< 5[122X and the remaining 2118 integers have [22X5≤
  q(n)≤ 54[122X; e.g. [22Xn=2^2.3.5.7^2.13[122X has [22Xq(n)=1221/23[122X.[133X
  
  
  [1X2.4 [33X[0;0YAn example session[133X[101X
  
  [33X[0;0YIn  this  section  we  outline  some examples of applications of the methods
  described  above.  We  included  runtimes  for all examples, but omitted the
  output in some cases, since it would be too long to be printed. The runtimes
  have  been obtained on an Intel(R) Pentium(R) 4 CPU 3.00GHz PC running under
  Linux.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:=5^2*7*13^2*67^2*97*107;[127X[104X
    [4X[28X1377938614325[128X[104X
    [4X[25Xgap>[125X [27XCubefreeOrderInfo(n,false);[127X[104X
    [4X[28X12[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFGroups(n));time;[127X[104X
    [4X[28X12[128X[104X
    [4X[28X53111[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:=19^2*23^2*29*37*73^2*107^2;[127X[104X
    [4X[28X12501895704027377[128X[104X
    [4X[25Xgap>[125X [27XCubefreeOrderInfo(n,false);[127X[104X
    [4X[28X24[128X[104X
    [4X[25Xgap>[125X [27XNumberCFGroups(n);time;[127X[104X
    [4X[28X24[128X[104X
    [4X[28X190536[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFGroups(n));time;[127X[104X
    [4X[28X24[128X[104X
    [4X[28X948319[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:=5^2*13*23^2*43^2*191;[127X[104X
    [4X[28X60716861075[128X[104X
    [4X[25Xgap>[125X [27XCubefreeOrderInfo(n,false);[127X[104X
    [4X[28X16[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFGroups(n)); time;[127X[104X
    [4X[28X16[128X[104X
    [4X[28X29146[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we compute some more data.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X n:=2*2*3*11*17*67;[127X[104X
    [4X[28X150348[128X[104X
    [4X[25Xgap>[125X [27XCubefreeOrderInfo(n,false);[127X[104X
    [4X[28X20[128X[104X
    [4X[25Xgap>[125X [27XNumberCFGroups(n);time;[127X[104X
    [4X[28X145[128X[104X
    [4X[28X12073[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFGroups(n)); time;[127X[104X
    [4X[28X145[128X[104X
    [4X[28X20757[128X[104X
    [4X[25Xgap>[125X [27XNumberCFSolvableGroups(n);time;[127X[104X
    [4X[28X144[128X[104X
    [4X[28X11925[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFSolvableGroups(n)); time;[127X[104X
    [4X[28X144[128X[104X
    [4X[28X18893[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFFrattiniFreeGroups(n)); time;[127X[104X
    [4X[28X109[128X[104X
    [4X[28X14421[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFNilpotentGroups(n));time;[127X[104X
    [4X[28X2[128X[104X
    [4X[28X12[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFSimpleGroups(n));time;[127X[104X
    [4X[28X1[128X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe consider another example with some info class output.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSetInfoLevel(InfoCF,1);[127X[104X
    [4X[25Xgap>[125X [27XConstructAllCFGroups(4620);;time;[127X[104X
    [4X[28X#I  Construct all groups of order 4620.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 2310.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 4620.[128X[104X
    [4X[28X#I  Construct 138 Frattini extensions.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 77.[128X[104X
    [4X[28X#I  Construct 1 Frattini extensions.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 7.[128X[104X
    [4X[28X#I  Construct 1 Frattini extensions.[128X[104X
    [4X[28X15501[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:=101^2*97*37^2*29^2;[127X[104X
    [4X[28X1139236591513[128X[104X
    [4X[25Xgap>[125X [27XCubefreeOrderInfo(n,false);[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XNumberCFGroups(n);time;[127X[104X
    [4X[28X8[128X[104X
    [4X[28X36[128X[104X
    [4X[25Xgap>[125X [27XSetInfoLevel(InfoCF,1);[127X[104X
    [4X[25Xgap>[125X [27XConstructAllCFGroups(n);time;[127X[104X
    [4X[28X#I  Construct all groups of order 1139236591513.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 10512181.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 304853249.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 388950697.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 1061730281.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 11279570213.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 30790178149.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 39284020397.[128X[104X
    [4X[28X#I    Compute solvable Frattini-free groups of order 1139236591513.[128X[104X
    [4X[28X#I  Construct 8 Frattini extensions.[128X[104X
    [4X[28X[ <pc group of size 1139236591513 with 7 generators>, [128X[104X
    [4X[28X  <pc group of size 1139236591513 with 7 generators>, [128X[104X
    [4X[28X  <pc group of size 1139236591513 with 7 generators>, [128X[104X
    [4X[28X  <pc group of size 1139236591513 with 7 generators>, [128X[104X
    [4X[28X  <pc group of size 1139236591513 with 7 generators>, [128X[104X
    [4X[28X  <pc group of size 1139236591513 with 7 generators>, [128X[104X
    [4X[28X  <pc group of size 1139236591513 with 7 generators>, [128X[104X
    [4X[28X  <pc group of size 1139236591513 with 7 generators> ][128X[104X
    [4X[28X1848[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  last example considers the cubefree order less than 50000 for which the
  number  of groups with this order is maximal: there are 3093 groups of order
  44100.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:=2*2*3*3*5*5*7*7;[127X[104X
    [4X[28X44100[128X[104X
    [4X[25Xgap>[125X [27XCubefreeOrderInfo(n,false);[127X[104X
    [4X[28X100[128X[104X
    [4X[25Xgap>[125X [27XNumberCFSolvableGroups(n,false);time;[127X[104X
    [4X[28X3087[128X[104X
    [4X[28X572639[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFSolvableGroups(n)); time;[127X[104X
    [4X[28X3087[128X[104X
    [4X[28X843085[128X[104X
    [4X[25Xgap>[125X [27XNumberCFGroups(n,false);time;[127X[104X
    [4X[28X3093[128X[104X
    [4X[28X719245[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFGroups(n)); time;[127X[104X
    [4X[28X3093[128X[104X
    [4X[28X1016763[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFFrattiniFreeGroups(n)); time;[127X[104X
    [4X[28X1305[128X[104X
    [4X[28X504451[128X[104X
    [4X[25Xgap>[125X [27XLength(ConstructAllCFNilpotentGroups(n));time;[127X[104X
    [4X[28X16[128X[104X
    [4X[28X180[128X[104X
  [4X[32X[104X
  
  
  [1X2.5 [33X[0;0YAccuracy check[133X[101X
  
  [33X[0;0YWe  have  compared  the  results of [10XConstructAllCFGroups[110X with the library of
  cubefree  groups  of  [5XSmallGroups[105X.  Further, we compared the solvable groups
  constructed  by  [10XIrreducibleSubgroupsOfGL[110X  with  the library of [5XIrredSol[105X. We
  have also done random isomorphism tests to verify that the list of groups we
  computed is not redundant.[133X
  
  [33X[0;0YOne  can call the following test files. The first one constructs some groups
  of order at most 2000 and compares the results with the [5XSmallGroups[105X library:[133X
  
  [33X[0;0Y[10XRereadPackage("cubefree","tst/testQuick.g");[110X[133X
  
  [33X[0;0YThe command[133X
  
  [33X[0;0Y[10XRereadPackage("cubefree","tst/testBig.g");[110X[133X
  
  [33X[0;0Yconstructs  the  solvable  groups  of a random cubefree (but not squarefree)
  order  at  most  [22X2^28-1[122X and does a random isomorphism test. Depending on the
  chosen number, the computation might not terminate due to memory problems.[133X
  
  [33X[0;0YThe  following  constructs  the  groups of three random cubefree orders less
  than  50000  compares  the result with the [5XSmallGroups[105X library. Depending on
  the chosen orders, this may take a while:[133X
  
  [33X[0;0Y[10XRereadPackage("cubefree","tst/testSG.g");[110X[133X
  
  [33X[0;0YThe  test  file [3XtestSGlong.g[103X constructs all cubefree groups of order at most
  50000  compares  the  results  with the [5XSmallGroups[105X library. There will be a
  positive  progress  report  every  50th order so that you can abort the test
  whenever you want.[133X
  
  [33X[0;0Y[10XRereadPackage("cubefree","tst/testSGlong.g");[110X[133X
  
  [33X[0;0YThree  of  these  four  test  files  use the function [10XCubefreeTestOrder[110X, see
  Section [14X2.1[114X.[133X
  
  [33X[0;0YThe  last  test  file compares some results of [10XIrreducibleSubgroupsOfGL[110X with
  the database of [5XIrredSol[105X. This may take a while:[133X
  
  [33X[0;0Y[10XRereadPackage("cubefree","tst/testMat.g");[110X[133X
  
