Medial Symmetry Type Graphs

A k-orbit map is a map with its automorphism group partitioning the set of flags into k orbits. Recently k-orbit maps were studied by Orbanić, Pellicer and Weiss, for k 6 4. In this paper we use symmetry type graphs to extend such study and classify all the types of 5-orbit maps, as well as all self-dual, properly and improperly, symmetry type of k-orbit maps with k 6 7. Moreover, we determine, for small values of k, all types of k-orbits maps that are medial maps. Self-dualities constitute an important tool in this quest.