Vertex-transitive maps on a torus

We examine FVT (free, vertex transitive) actions of wallpaper groups on semiregular tilings. By taking quotients by lattices we then obtain various families of FVT maps on a torus, and describe the presentations of groups acting on the torus. Altogether there are 29 families, 5 arising from the orientation preserving wallpaper groups and 2 from each of the remaining wallpaper groups. We prove that all vertex-transitive maps on torus admit an FVT map structure.