# The Genus-C^{6} Regular Map {3,10}_{5}

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

Its dual is C^{6}:{10,3}_{5}.

Its rotational symmetry group is S5.

Its Petrie polygons have five edges.
It is the Petrie dual of the dodecahedron S^{0}:{5,3}.

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

### Antipodal Faces and Vertices

The faces form antipodal pairs (shown in the same colour in the diagram).
The edges form antipodal pairs.
Each vertex is antipodal to a Petrie polygon.