t is well known that a regular Belyi function B : S -> P^1(C) on a surface S determines a regular dessin D on S and a realization of the surface as quotient S = H/U of the hyperbolic plane. The group U is a normal, torsion free subgroup of a triangle group T. The group G = T/U is a group of automorphisms of S, specifically the group of covering transformation of B. Also, the surface S has defining equations with coefficients in a number field K. An element of the absolute Galois group determines a Galois conjugate surface S' by acting on the coefficients of a defining equation of S. There is an associated dessin D', and subgroup U' contained in T such that S' = H/U'. The goal of this talk is to establish as much as possible about D' and U' from the knowledge of the pair T,U. In certain well-known examples the dessin D' is obtained by Wilson operations. Here we also consider construction of the dessin when Wilson operations are not valid. We illustrate the methods with low genus surfaces and cyclic n-gonal surfaces.