GrpConst

Constructing the Groups of a Given Order

Version 2.6.1
Released 2018-08-09

This project is maintained by Bettina Eick, Max Horn

README file for the GrpConst share package by
Hans Ulrich Besche and Bettina Eick.

The package contains programs that implement three different approaches to
constructing up to isomorphism all groups of a given order.

The FrattiniExtensionMethod constructs all soluble groups of a given order.
On request it gives only those that are (or are not) nilpotent or
supersolvable or that do (or do not) have normal Sylow subgroups for
some given set of primes. The program's output may be expressed in a
compact coded form, if desired.

The CyclicSplitExtensionMethod constructs all (necessarily soluble) groups
whose given orders are of the form p^n*q for different primes p and q and
which have at least one normal Sylow subgroup. The method, which relies
upon having available a list of all groups of order p^n, is often faster
than the Frattini extension method for the groups to which it applies.

The UpwardsExtensions takes as its input a permutation group G and positive
integer s and returns a list of permutation groups, one for each
extension of G by a soluble group of order a divisor of s. Usually it is
used for nonsoluble G only, since for soluble groups the above methods
are more efficient.

The programs in this package have been used to construct a large part
of the Small Groups library. The algorithms upon which they are based
are original work of the package authors and are described fully in

[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. U. Besche and B. Eick.
The groups of order $q^n \cdot p$,
Comm Algebra. {\bf 29} (2001), 1759 -- 1772.