Construction of Large Permutation Representations for Matrix Groups

This article describes the general computational tools for a new proof of the existence of the large sporadic simple Janko group J₄ [10] given by Cooperman, Lempken, Michler and the author [7] which is independent of Norton [12] and Benson [1].Its basic step requires a generalization of the Cooperman, Finkelstein, Tselman and York algorithm [6] transforming a matrix group into a permutation group.An efficient implementation of this algorithm on high performance parallel computers is described. Another general algorithm is given for the construction of representatives of the double cosets of the stabilizer of this permutation representation. It is then used to compute a base and strong generating set for the permutation group. In particular, we obtain an algorithm for computing the group order of a large matrix subgroup of GL _n (q), provided we are given enough computational means.It is applied to the subgroup G = 〈x, y〉 < GL₁₃₃₃(11) corresponding to Lempken’s construction of J₄ [11].