Medial symmetry type graphs

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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 Orbani c, Pellicer and Weiss, for k ≤ 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≤ 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.