HN has an involution whose
centralizer is of the form 2.HS.2, where HS is the
Higman-Sims group (which is how Harada found it).
The prime 5 plays a special role in the group. For example, it centralizes an element of order 5 in the
Monster group (which is how Norton found it), and as a result acts naturally on a
vertex operator algebra over the field with 5 elements (
Lux, Noeske & Ryba 2008). This implies that it acts on a 133 dimensional algebra over F5 with a commutative but nonassociative product, analogous to the
Griess algebra (
Ryba 1996).
The full nomralizer of a 5A element in the Monster group is (D10 × HN).2, so HN centralizes 5 involutions alongside the 5-cycle. These involutions are centralized by the
Baby monster group, which therefore contains HN as a subgroup.
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.
To recall, the prime number 5 plays a special role in the group and for HN, the relevant McKay-Thompson series is where one can set the constant term a(0) = −6 (OEIS:
A007251),
M12:2 (Two classes, fused by an outer automorphism)
34:2.(A4 × A4).4
31+4:4.A5
References
Harada, Koichiro (1976), "On the simple group F of order 214 · 36 · 56 · 7 · 11 · 19", Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975), Boston, MA:
Academic Press, pp. 119–276,
MR0401904
Ryba, Alexander J. E. (1996), "A natural invariant algebra for the Harada-Norton group", Mathematical Proceedings of the Cambridge Philosophical Society, 119 (4): 597–614,
doi:
10.1017/S0305004100074454,
ISSN0305-0041,
MR1362942