A basic set generates an abelian group of order 210, which extends in Fi22 to a subgroup 210:M22.
Next one takes Fi23 and treats it as a one-point stabilizer for a graph of 306936 (= 23⋅33⋅72⋅29) vertices to make a group Fi24.
Fischer's notation for them was M(22), M(23) and M(24)′, which emphasised their close relationship with the three largest Mathieu groups, M22, M23 and M24.
John H. Conway and Simon P. 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 (main or principal moduli) from simple combinations of dimensions of sporadic groups.