The transitive groups of degree twelve
The author describes an elementary method for constructing all the pairwise nonconjugate transitive groups of degree 12. The method has been implemented by the author using the group theory package CAYLEY, yielding a list of 301 transitive groups of degree 12. Details are also given of the 17 groups in this list which are minimal transitive. The author compares his results with those of \textit{G. A. Miller} [Q. J. Math. 28, 193-231 (1896; JFM 27.0097.03)], who also gave a list of transitive groups of degree 12. Miller's list differs from the author's in surprisingly few places. The basic method is as follows. Let \(G\) be a transitive group of degree 12. If \(G\) is primitive then \(G\) is one of the groups \(PSL_ 2(11)\), \(PGL_ 2(11)\), \(M_{11}\), \(M_{12}\), \(A_{12}\) or \(S_{12}\) [see \textit{C. C. Sims}, in Comput. Probl. abstract Algebra, Oxford 1967, 169-183 (1970; Zbl 0215.10002)], so one may suppose that \(G\) is imprimitive, with a block system consisting of \(c\) blocks of size \(d\), where \(cd=12\). The various values of \(c\) are dealt with case by case, the method being to identify the possibilities for the kernel \(K\) of the action of \(G\) on the block system. Then \(G/K\leq N(K)/K\), and the group \(N(K)/K\) is usually small enough for the subgroup lattice programme in CAYLEY to be applied to give a list of possibilities for \(G/K\), and hence for \(G\).
scientific article; zbMATH DE number 4118567