# The Genus-C^{6} Regular Map {10,3}_{10}

This regular map is in genus-C^{6} (a sphere plus six
crosscaps). It has six decagonal faces, each meeting each of
the others twice. It has 20 vertices and 30 edges, giving a
Euler characteristic of -4.

Its graph is the
Desargues graph.

Its dual is C^{6}:{3,10}.

Its rotational symmetry group is S5.

Its Petrie polygons have 10 edges. It is its own Petrie dual.

**It is different from C**^{6}:{10,3}_{5}.
C^{6}:{10,3}_{10} has a girth (minimal loop) of six edges,
C^{6}:{10,3}_{5} has a girth of five.

### Antipodal Faces and Vertices

The vertices form antipodal pairs.
The edges form antipodal pairs.
Each face is antipodal to a Petrie polygon.

There are hexagons formed by traversing the edges and turning left, left, right, right,
left, left, right, right, etc., at successive vertices. Each such hexagon is antipodal
to another such hexagon and to two faces. The diagram to the left show an antipodal set
of two of these hexagons and two vertices.