Construction of finite groups
The authors give an algorithm to determine all groups of a given order up to isomorphism. Following a suggestion of \textit{W. Gaschütz} [Math. Z. 58, 160-170 (1953; Zbl 0050.02202)] they first construct the possible solvable Frattini factors \(G/\Phi(G)\). These factors are obtained as subdirect products of irreducible subgroups of \(\text{GL}(d,q)\). In a second step the factors are extended by modules in such a way that one obtains Frattini extensions (i.e., the Frattini factor group of the extension stays the same). This extension step follows \textit{W. Plesken} [J. Symb. Comput. 4, 111-122 (1987; Zbl 0635.20013)]. Isomorphisms among the resulting groups which preserve the extension structure are dealt with by taking only orbit representatives under the action of a symmetry group on the cohomology \(H^2(G/\Phi(G),M)\). To remove all further isomorphisms, the authors distinguish non-isomorphic groups by invariant properties (such as the power map) and search for isomorphisms between solvable groups by a random isomorphism test based on finding ``isomorphic'' special polycyclic generating systems (see \textit{B. Eick} [Groups and computation II, DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 28, 101-112 (1997; Zbl 0871.20016)]) for different groups. This approach adapts naturally towards the construction of solvable groups only within a given saturated formation. For the construction of nonsolvable groups the authors use upward cyclic extension as proposed by \textit{R. Laue} [Bayreuther Math. Schr. 9 (1982; Zbl 0479.20010)]. Section 5 describes how to get all groups of order \(p^nq\) from a list of the groups of order \(p^n\). The methods described have been used sucessfully by the authors [J. Symb. Comp. 27, No. 4, 405-413 (1999; see the following review Zbl 0922.20002)] for many small orders, an implementation is available as a share package for GAP.
scientific article; zbMATH DE number 1275078