Monster Group - Representations

Representations

The minimal degree of a faithful complex representation is 196883, which is the product of the 3 largest prime divisors of the order of M. The character table of the Monster, a 194-by-194 array, was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. The smallest linear representation over any field has dimension 196882 over the field with 2 elements, only 1 less than the dimension of the smallest complex representation.

The smallest faithful permutation representation of the monster is on 24 · 37 · 53 · 74 · 11 · 132 · 29 · 41 · 59 · 71 (about 1020) points.

The monster can be realized as a Galois group over the rational numbers (Thompson 1984, p. 443), and as a Hurwitz group (Wilson 2004).

The monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of "small" representations. For example, the simple groups A₁₀₀ and SL₂₀(2) are far larger, but easy to calculate with as they have "small" permutation or linear representations. The alternating groups have permutation representations that are "small" compared to the size of the group, and all finite simple groups of Lie type have linear representations that are "small" compared to the size of the group. All sporadic groups other than the monster also have linear representations small enough that they are easy to work with on a computer (the next hardest case after the monster is the baby monster, with a representation of dimension 4370).