Constructing groups of a given order
This package contains GAP implementations of 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 method 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.