HS is one of the 26 sporadic groups and was found by Donald G. Higman and Charles C. Sims (1968).
Inspired by this they decided to check for other rank 3 permutation groups on 100 points.
By putting together these two representations, they found HS, with a one-point stabilizer isomorphic to M22.
HS is the simple subgroup of index two in the group of automorphisms of the Higman–Sims graph.
In Co0 HS arises as a pointwise stabilizer of a 2-3-3 triangle, one whose edges (differences of vertices) are type 2 and 3 vectors.
In fact, |HS| = 100 |M22|, and there are instances of HS including a permutation matrix representation of the Mathieu group M22.
Magliveras (1971) found the 12 conjugacy classes of maximal subgroups of HS as follows:
[4] Listed are 2 permutation representations: on the 100 vertices of the Higman–Sims graph, and on the 176 points of Graham Higman's geometry.
Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups.