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