There are many methods of constructing an explicit representation of a given group, starting from only theoretical information about the group such as characters and subgroup structure. These range from (a) constructing a set of generators and relations from the subgroups, and then using Todd-Coxeter to make a permutation representation, to (b) making matrix representations of subgroups and gluing them together to make a matrix representation of the whole group.

I shall concentrate on the latter, illustrating with a 3-dimensional example the basic ideas that apply even to making a 196882-dimensional representation of a group of order 808017424794512875886459904961710757005754368000000000.