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.