The Genus-C⁶ Map {6,5}

The figures shown and described here are not regular maps. For the one on the left, one Petrie polygons has 15 edges, one has ten, and one has five, so it cannot be edge-transitive. For the one on the right, each vertex is connected to one other vertex once and to two others twice each, so it cannot be edge-transitive.

These maps are in genus-C⁶ (a sphere plus six crosscaps). They have five pentagonal faces, all meeting at each of the six five-valent vertices. They have 15 edges, giving a Euler characteristic of -4.

Their duals are shown at C⁶:{5,6}.

If the central vertex in the diagram, and its five edges, are removed from the one on the left, we get the regular map C⁶:{20,4}. If we do the same to the one on the right, we get an irregular C⁶:{20,4}