9. Further Examples

In the first part of this chapter we outline the classification of the regular embeddings of two families of regular graphs, namely certain generalised Paley graphs (using Cayley maps for finite fields), and the complete bipartite graphs K _n, n . We describe their automorphism groups, and characterise the generalised Paley maps as those maps for which the automorphism group acts primitively and faithfully on the vertices. In the case of the complete bipartite graphs, results of Huppert, Itô and Wielandt on factorisations of groups, and of Hall on solvable groups, are used in the classification. We show how Wilson operations act on these two families of maps, and we use this to investigate their Galois orbits and fields of definition. In the second part of this chapter we extend the action of Wilson operations from regular maps to regular dessins, concentrating on those dessins which embed a complete bipartite graph K _p, q where p and q are distinct primes. Under suitably favourable conditions on p and q, we can classify these dessins and use Wilson operations to determine their Galois orbits and fields of definition. Finally, we determine explicit equations for the associated quasiplatonic curves, a problem which is completely intractable in most other cases.