THE GROUPS OF ORDER qⁿ· p

Abstract

We describe a method to determine up to isomorphism the groups of order q n · p for a fixed prime-power q n and indeterminate prime p = q. We report on the explicit construction of all groups of order 2 n · p for n ≤ 8 and 3 n · p for n ≤ 6. In particular, we show that there are 1 090 235 groups of order 768.

An Atlas of Small Regular Abstract Polytopes

“…The algorithm of Section 2 was implemented in GAP Version 4.4 (see [5]). It was applied to numerous groups from the GAP Library of Small Groups, developed by Besche, Eick and others (see [3], which builds on [1] and [2]). The groups were converted to permutation groups (this seemed to speed things up) and the polytopes for each group were extracted using the algorithm and stored.…”

The construction of finite solvable groups revisited

“…Then Aut(A) = GL(n, 2). The relevant subgroups U of Aut(A) are the solvable subgroups of GL(n, 2) of order 2 (1,3). The relevant subgroups of Aut(A) are the solvable subgroups U of Aut(A) of order 2 5−n with O(U ) = {1}.…”

“…Nilpotent groups are determined as direct products of p-groups and p-groups can be constructed using the p-group generation algorithm [21]. Solvable groups can be determined by the Frattini extension method [2] or the cyclic split extensions methods [3]. Non-solvable groups can be obtained via cyclic extensions of perfect groups as in [2] or via the method in [1].…”