Generalized Cayley maps and their Petrie duals

Cayley maps are embeddings of Cayley graphs in orientable surfaces which possess a group of orientation preserving automorphisms acting regularly on the vertices. We generalize the concept of a Cayley map by considering embeddings of Cayley graphs in both orientable and non-orientable surfaces and by requiring a group of automorphisms acting regularly on vertices that does not have to consist entirely of orientation preserving automorphisms. This leads to new families of maps in both the orientable and non-orientable cases. Since the Petrie dual operator preserves the property of being a generalized Cayley map, throughout the paper we consider the action of this operator on our maps.