README file for the Cubefree Package by Heiko Dietrich.

This package contains an implementation of an algorithm to construct up 
to isomorphism all groups of a given cubefree order. The algorithm is
based on the ideas in [1] and [2] and it is fully described in [3].

The algorithm needs a method to construct all irreducible subgroups of
GL(2,p) up to conjugacy. We use the method described in [4] for this
purpose. In turn, the algorithm of [4] requires a method for writing
an irreducible matrix groups over a minimal finite field. We use the
algorithm described in [5] for this purpose.

The main functions of the package are:

(1) ConstructAllCFGroups( n ) 
... constructs all  groups of a given cubefree order n.

(2) ConstructAllCFSimpleGroups( n ) 
... constructs all simple groups of a given cubefree order n.

(3) ConstructAllCFNilpotentGroups( n ) 
... constructs all nilpotent groups of a given cubefree order n.

(4) ConstructAllCFFrattiniFreeGroups( n ) 
... constructs all Frattini-free groups of a given cubefree order n.

(5) 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.

(6) NumberCFGroups( n )
... returns the number of all cubefree groups of order n.

(7) NumberCFSolvableGroups( n )
... returns the number of all cubefree solvable groups of order n.

(8) 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.

(9) RewriteAbsolutelyIrreducibleMatrixGroup( G ) 
... rewrites an absolutely irreducible subgroup G\leq GL(n,q) over the
subfield generated by the traces of the elements of G.


References:

[1] H. U. Besche and B. Eick.
    Construction of finite groups,
    J. Symb. Comput. {\bf 27} (1999), 387 -- 404.

[2] H. U. Besche and B. Eick.
    The groups of order at most 1000 except 512 and 768,
    J. Symb. Comput. {\bf 27} (1999), 405 -- 413.

[3] H. Dietrich and B. Eick.
    On The Groups Of Cube-Free Order
    J. Algebra {\bf 292} (2005), 122 -- 137

[4] D. L. Flannery and E. A. O'Brien.
    Linear Groups of small degree over finite fields,
    Intern. J. Alg. Comput. {\bf 15] (2005), 467 -- 502

[5] S. P. Glasby and R. B. Howlett.
    Writing representations over minimal fields,
    Comm. Alg. {\bf 25}(6) (1997) 1703 -- 1711.

