Monstrous Moonshine

The Monster group’s symmetries number about 8×10⁵³ — far too many to list. But up to change of basis there are exactly 194 types of symmetry, its conjugacy classes. This is the pastable dictionary of all of them, each paired with its McKay–Thompson series — the modular function the moonshine module V♮ assigns to it, named by the genus-zero group it is a Hauptmodul of. 194 classes → 171 distinct series.

→ New here? Start with the visual explainer: why these numbers matter and what the groups do

Conjugacy classes

194

Distinct series

171

Element orders

|Monster|

…

Class 1A → the j-function. The identity’s McKay–Thompson series is J = j − 744, whose coefficients are the graded dimensions of V♮:

Class	Order	McKay–Thompson series Tɡ · moonshine group

Reading the group symbols

Every one of these groups is genus zero — that is the content of the moonshine theorem (conjectured by Conway–Norton, proved by Borcherds). The 15 primes whose Fricke group Γ₀(p)+ is genus zero are exactly the supersingular primes — the prime divisors of |Monster| (Ogg’s observation, the “Jack Daniel’s problem”). Their prime-order classes are tagged Ogg.