Computing in the Monster

We give a survey of computational methods and results concerning the Monster sporadic simple group. There are now three computer constructions of the Monster which are proving effective in answering real questions about this group. The first construction over the field of two elements is the fastest for calculations, and has been used to show the group is a Hurwitz group. The second construction over the field of three elements, uses an involution centralizer as the heart of the construction, and has proved to be the most useful as far as calculations with subgroups is concerned. P. E. Holmes has used this construction to find explicitly four new conjugacy classes of maximal subgroups, as well as to eliminate various other possibilities for maximal subgroups. The third construction over the field of seven elements uses the same generators as the first construction, which means that elements given as words in these generators can be investigated modulo 2 and modulo 7 simultaneously. This gives enough information in most cases to determine the conjugacy class of the element.