  A map is a connected topological graph $\Gamma$ cellularly embedded in a
surface. In this paper, applying Tutte's algebraic representation of map, new
ideas for enumerating non-equivalent orientable or non-orientable maps of graph
are presented. By determining automorphisms of maps of Cayley graph
$\Gamma={\rm Cay}(G:S)$ with ${\rm Aut} \Gamma\cong G\times H$ on locally,
orientable and non-orientable surfaces, formulae for the number of
non-equivalent maps of $\Gamma$ on surfaces (orientable, non-orientable or
locally orientable) are obtained . Meanwhile, using reseults on GRR graph for
finite groups, we enumerate the non-equivalent maps of GRR graph of symmetric
groups, groups generated by 3 involutions and abelian groups on orientable or
non-orientable surfaces.
