group with the same centralizers of involutions as the monster is isomorphic to the monster). The monster was a culmination of the development of sporadic simple groups and can be built from any two of three subquotients: the Fischer group Fi24, the baby monster, and the Conway group Co1. The Schur multiplier and the outer automorphism group of the monster are both trivial. Representations The minimal degree of a faithful complex representation is 47 × 59 × 71 = 196,883, hence is the product of the three largest prime divisors of the order of M. The smallest faithful linear representation over any field has dimension 196,882 over the field with two elements, only one less than the dimension of the smallest faithful 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, and as a Hurwitz group. 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. Alternating groups, such as A100, have permutation representations that are "small" compared to the size of the group, and all finite simple groups of Lie type, such as SL20(2), 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). A computer construction Robert A. Wilson has found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements in the field of order 2) which together generate the monster group by matrix multiplication; this is one dimension lower than the 196,883-dimensional representation in characteristic 0. Performing calculations with these matrices is possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes. Wilson asserts that