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.
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.
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