The Held group is the third possibility, and its construction was completed by John McKay and Graham Higman.
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.
where one can set the constant term a(0) = 10 (OEIS: A007264), and η(τ) is the Dedekind eta function.
It can be defined in terms of the generators a and b and relations Butler (1981) found the 11 conjugacy classes of maximal subgroups of He as follows: