Faithful permutation representations of toroidal regular maps

TL;DR

This paper classifies all degrees of faithful transitive permutation representations of symmetry groups of certain toroidal regular maps and provides example graphs called CPR-graphs for some cases.

Contribution

It systematically enumerates all possible degrees of faithful permutation representations for these groups and introduces CPR-graphs as concrete examples.

Findings

Complete list of degrees for faithful transitive representations of the groups.

Construction of CPR-graphs illustrating some permutation representations.

Insights into the symmetry structures of toroidal regular maps.

Abstract

In this paper we list all possible degrees of a faithful transitive permutation representation of the group of symmetries of a regular map of types $\{4,4\}$ and $\{3,6\}$ and we give examples of graphs, called CPR-graphs, representing some of these permutation representations.