### What is the smallest pair of regular maps in the same genus and having the same Schläfli symbol and the same number of edges?

In the torus, there are two regular maps with the Schläfli symbol
{4,4}, 25 faces, 25 vertices, and 50 edges. They are
{4,4}_{(5,0)}
and
{4,4}_{(4,3)}.
They can be distinguished easily: the latter has bigger Petrie polygons,
bigger holes, and is chiral.

The smallest example for which neither is chiral is in S3.
They have Schläfli symbol {8,8}, and 28 edges. They are
S3:{8,8}_{4}
and
S3:{8,8}_{2}.
The former has Petrie polygons with four edges; the latter, with two.

Regular Maps FAQ to which this is one of the answers.