The Genus-S⁷ Regular Map {6,9}

This regular map has nine hexagonal faces, six vertices, and 27 edges.

Its rotational symmetry group has order 54.

Its Petrie polygons have 18 edges, its holes have six, its order-2 Petrie polygons have 18, its order-3 holes have two, its order-3 Petrie polygons have six edges (three each counted once in each direction), its order-4 holes have six, and its order-3 Petrie polygons have 18.

It is chiral. Apart from regular maps in the torus, it is the smallest chiral regular map, both in number of edges, and in genus (there is one other chiral regular map of genus 7, with 28 edges).