Construction of regular maps from their small quotients

Every bicolored map may be represented by a triple of permutations (x,y,z) acting on the set E of edges and such that xyz=1. Here the cycles of x are black vertices, the cycles of y are white vertices, and the cycles of z are faces. To every map one can associate two groups: the monodromy group G=<x,y,z>, and the automorphism group H. A map is called regular if these two groups are isomorphic. In this case the set E of edges can be identified with the group, and this group acts on itself by multiplications. Thus, a construction of a regular map, even a large one, may be reduces to a construction of a group with desired properties, and this group may be constructed as a monodromy group of another map, often much smaller.
As an example of special interest we will consider Hurwitz maps. In 1893, Hurwitz proved that for a map of genus g>1 the order of its automorphism group is bounded by 84(g-1). Hurwitz maps are interesting not only because they are very symmetric but also because they are very rare. Marston Conder (Aucland) classified all regular maps of genus from 2 to 101. Their number is more 19 thousand, and only seven of them are Hurwitz.
This is a joint work with Gareth Jones (Southampton).