196,883 dimensions and the Monster

The least number of dimensions in which the Monster group can act non-trivially is 196,883. This number is 47.59.71, the product of the three largest prime numbers dividing the size of the Monster, but its main point of interest is that by adding 1 we obtain 196,884. This is the first non-trivial number appearing in an important sequence in number theory, a fact first observed by John McKay. It led later to the Moonshine phenomena, as explained in my book Symmetry and the Monster.

The initial stages of the Moonshine investigation looked at all the irreducible representations of the Monster. There are 194 of these: one is the trivial 1-dimensional representation where the group does nothing at all; the next smallest has dimension 196,883; and the next smallest after that has dimension 21,296,876. Using these numbers, and others, John Thompson found that McKay's observation was part of a larger pattern, and his work inspired John Conway and Simon Norton to develop the ideas that led to their celebrated Monstrous Moonshine paper.

A few years later, Robert Griess constructed the Monster in 196,884 dimensions by creating an algebra structure, and showing that it preserved this structure. Griess's representation is a very natural one, but is not irreducible — it splits into two: the trivial 1-dimensional representation, and the irreducible representation in 196,883 dimensions.