A simple group of order 44,352,000

Abstract

The group G of the title is obtained as a primitive permutation group of degree 100 in which the stabilizer of a point has orbits of lengths 1, 22 and 77 and is isomorphic to the Mathieu group M22. Thus G has rank 3 in the sense of [1]. G is an automorphism group of a graph constructed from the Steiner system ~ (3, 6, 22). WITT [3] defined a Steiner system ~(d, rn, n) to be a set S of n points together with a set B of subsets of S (referred to here as blocks) such that each block contains exactly m points and each set of d points is contained in exactly one block. WITT [4] showed that Steiner systems ~ (3, 6, 22) exist and that they are unique up to isomorphism. The automorphism group Mz2 of an ~ (3, 6, 22) contains the Mathieu group Mz2 as a subgroup of index 2 and is the normalizer of M22 in M24. Throughout the rest of the paper we shall use the following notation: S and B will denote the sets of points and blocks, respectively, of a fixed ~(3, 6, 22). Points will be denoted by Greek letters ~, fl, ... and blocks by Roman letters u, v, .... For each o~eS, [~] will denote the set of blocks containing ~. We shall use the following facts about ~(3, 6, 22) and M22: