# Higman–Sims group

In the mathematical field of group theory, the Higman–Sims group HS is a sporadic simple group found by Donald G. Higman and Charles C. Sims (1968) of order

29 · 32 · 53 · 7 · 11
= 44352000.
≈ 4 · 107.

It is the simple subgroup of index two in the group of automorphisms of the Higman–Sims graph. The Higman–Sims graph has 100 nodes, so the Higman–Sims group HS is a transitive group of permutations of a 100 element set.

The Higman–Sims group was discovered in 1967, when Higman and Sims were attending a presentation by Marshall Hall on the Hall–Janko group. This is also a permutation group of 100 points, and the stabilizer of a point is a subgroup with two other orbits of lengths 36 and 63. Inspired by this they decided to check for other rank 3 permutation groups on 100 points. They soon focused on a possible one containing the Mathieu group M22, which has permutation representations on 22 and 77 points. (The latter representation arises because the M22 Steiner system has 77 blocks.) By putting together these two representations, they found HS, with a one-point stabilizer isomorphic to M22.

Graham Higman (1969) independently discovered the group as a doubly transitive permutation group acting on a certain 'geometry' on 176 points.

The Schur multiplier has order 2, the outer automorphism group has order 2, and the group 2.HS.2 appears as an involution centralizer in the Harada–Norton group.

## Generalized Monstrous Moonshine

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For HS, the McKay-Thompson series is $T_{10A}(\tau)$ where one can set a(0) = 4 (),

\begin{align}j_{10A}(\tau) &=T_{10A}(\tau)+4\\ &=\Big(\big(\tfrac{\eta(\tau)\,\eta(5\tau)}{\eta(2\tau)\,\eta(10\tau)}\big)^{2}+2^2 \big(\tfrac{\eta(2\tau)\,\eta(10\tau)}{\eta(\tau)\,\eta(5\tau)}\big)^{2}\Big)^2\\ &=\Big(\big(\tfrac{\eta(\tau)\,\eta(2\tau)}{\eta(5\tau)\,\eta(10\tau)}\big)+5 \big(\tfrac{\eta(5\tau)\,\eta(10\tau)}{\eta(\tau)\,\eta(2\tau)}\big)\Big)^2-4\\ &=\frac{1}{q} + 4 + 22q + 56q^2 +177q^3+352q^4+870q^5+1584q^6+\dots \end{align}

Let $r_n$ = 1, 22, 77, 154, 175, 231, 693, 770, 825, 896, 1056, 1386, 1408,...() be the degrees of irreducible representations of HS. Then,

\begin{align} 1 &= r_1\\ 22 &= r_2\\ 56&= r_1-r_2+r_3\\ 177&= r_1 + r_2 + r_4\\ 352&= r_1 + r_2 + r_4+r_5\\ 870&= r_1 + r_2 + r_4+r_7\\ 1584&= r_1+r_5+r_{13}\\ \end{align}

and so on. For a specific value,

$j_{10A}\Big(\tfrac{5+\sqrt{-85}}{10}\Big) = -18^2$

## Relationship with the Conway groups

Conway (1968) showed how the Higman–Sims graph could be embedded in the Leech lattice. Here, HS fixes a 2-3-3 triangle and a 22-dimensional sublattice. The Higman–Sims group thus becomes a subgroup of each of the Conway groups Co0, Co2 and Co3. If a conjugate of HS in Co0 fixes a particular point of type 3, this point is found in 276 triangles of type 2-2-3, which this copy of HS permutes in orbits of 176 and 100. This also provides a 22 dimensional representation of HS, acting on a 22 dimensional lattice given by the orthogonal complement of a 2-3-3 triangle with one vertex at the origin.

## Maximal subgroups

Magliveras (1971) showed that HS has 12 conjugacy classes of maximal subgroups.

• M22, order 443520
• U3(5):2, order 252000 – one-point stabilizer in doubly transitive representation of degree 176
• U3(5):2 – conjugate to class above in HS:2
• PSL(3,4):2, order 40320
• S8, order 40320
• 24.S6, order 11520
• 43:PSL(3,2), order 10752
• M11, order 7920
• M11 – conjugate to class above in HS:2
• 4.24.S5, order 7680 – centralizer of involution moving 80 vertices of Higman–Sims graph
• 2 × A6.22, order 2880 – centralizer of involution moving all 100 vertices
• 5:4 × A5, order 1200