The installation of the Cubefree package follows standard GAP rules.
So the standard method is to unpack the package into the
directory of your GAP distribution. This will create an
The package is written for GAP 4. It requires the packages GrpConst
and Polycyclic, and, if the test file
tst/testMat.g is called, the
This package contains an implementation of an algorithm to construct all groups of a (reasonable) given cubefree order up to isomorphism. This algorithm is based on the ideas in  and  and it is fully described in . The implementation needs a method to construct all irreducible subgroups of GL(2,p) up to conjugacy. We use the method described in  for this purpose. In turn, the algorithm of  requires a method for writing an irreducible matrix group over a minimal finite field. We use the algorithm described in  for this purpose.
The main functions of the package are the following. Please see the documentation for a more detailed description.
ConstructAllCFGroups( n )
… constructs all groups of a given cubefree order n.
ConstructAllCFSimpleGroups( n )
… constructs all simple groups of a given cubefree order n.
ConstructAllCFSolvableGroups( n )
… constructs all solvable groups of a given cubefree order n.
ConstructAllCFNilpotentGroups( n )
… constructs all nilpotent groups of a given cubefree order n.
ConstructAllCFFrattiniFreeGroups( n )
… constructs all Frattini-free groups of a given cubefree order n.
NumberCFGroups( n )
… returns the number of all groups of a given cubefree order n.
NumberCFSolvableGroups( n )
… returns the number of all solvable groups of given cubefree order n.
CountAllCFGroupUpTo( n )
… counts all cubefree groups of order at most n.
The output is a list L whose i.th entry is the number of groups
of order i up to isomorphism if i is cube-free and unbound, otherwise.
IrreducibleSubgroupsOfGL( 2, q )
… computes all irreducible subgroups of GL(2,q) up to conjugacy where q=p^r
is a prime-power with p>=5.
RewriteAbsolutelyIrreducibleMatrixGroup( G )
… rewrites an absolutely irreducible subgroup G\leq GL(n,q) over the
subfield generated by the traces of the elements of G.
CubefreeOrderInfo( n )
… displays some (very vague) information about the complexity
of the construction of the groups of (cubefree) order
CubefreeTestOrder( n )
… tests the functionality of the functions (1)–(7) and compares it with
the data of the SmallGroups library. It returns true if everything is okay,
otherwise an error message will be displayed.
For some input the counting functions use some data of the SmallGroups
library. This can be avoided, see the documentation of (6) and (7).
Moreover, in the case of squarefree orders and orders of the type p^2 or
p^2q it is more practical to use the functions
NumberSmallGroups of the
SmallGroups library; please see the documentation for
There are several test files to call to check the results of the standard
methods. The standard test file is
tst/testQuick.g. Moreover, there are
Please see the documentation for more information.
Heiko Dietrich firstname.lastname@example.org, Monash University
Issues can also be reported at https://github.com/gap-packages/cubefree/issues.
 H. U. Besche and B. Eick. Construction of finite groups, J. Symb. Comput. 27 (1999), 387 – 404.
 H. U. Besche and B. Eick. The groups of order at most 1000 except 512 and 768, J. Symb. Comput. 2 (1999), 405 – 413.
 H. Dietrich and B. Eick. On The Groups Of Cube-Free Order J. Algebra 292 (2005), 122 – 137
 D. L. Flannery and E. A. O`Brien. Linear Groups of small degree over finite fields, Intern. J. Alg. Comput. 15 (2005), 467 – 502
 S. P. Glasby and R. B. Howlett. Writing representations over minimal fields, Comm. Alg. 25(6) (1997) 1703 – 1711.