CONSTRUCTION OF Co1 FROM AN IRREDUCIBLE SUBGROUP M24 OF GL11(2)

In this article we give an self contained existence proof for J. Conway’s sporadic simple group Co1 [4] using the second author’s algorithm [14] constructing finite simple groups from irreducible subgroups of GLn(2). Here n = 11 and the irreducible subgroup is the Mathieu group M24. From the split extension E of M24 by a uniquely determined 11-dimensional GF(2)M24module V we construct the centralizer H = CG(z) of a 2-central involution z of E in an unknown target group G. Then we prove that all the conditions of Algorithm 2.5 of [14] are satisfied. This allows us to construct a simple subgroup G of GL276(23) which we prove to be isomorphic with Conway’s original sporadic simple group Co1 by means of a constructed faithful permutation representation of G and Soicher’s presentation [16] of the original Conway group