The Genus-2 Map {10,3}
The figure shown and described below is not a regular map.
See below.
This genus-2 map has three decagonal faces, all three meeting
at each of its ten vertices. It has 15 edges, and a Euler
characteristic of -2. It is shown to the right.
Its dual is {3,10}.
It appears to have a rotational symmetry group of order 30; but on examination
it is found not to be symmetrical. To see this lack of symmetry, label the edges,
arbitrarily, as shown to the left. Go round each of the decagons clockwise, noting
alternate edge-labels. We find the six cycles (ABCDE) (GHIJK) (PQRST) (KHJGI)
(PQRST) (ACEBD). The set of edges {P,Q,R,S,T} appears in the same cyclic order
twice, the other two sets of edges appear in two different cyclic orders.