The Degrees Of Toroidal Regular Proper Hypermaps

TL;DR

This paper completes the classification of symmetry groups of regular toroidal hypermaps of type (3,3,3) on genus 1 surfaces, focusing on subgroups of index 2 of the symmetry group of a specific toroidal map.

Contribution

It provides a detailed analysis of the subgroup structure related to regular toroidal hypermaps of type (3,3,3) on genus 1 surfaces, extending previous classifications.

Findings

Classification of subgroups of index 2 in the symmetry group of a toroidal map.

Identification of hypermaps of type (3,3,3) on genus 1 surfaces.

Completion of the symmetry classification for these hypermaps.

Abstract

Recently the classification of all possible faithful transitive permutation representations of the group of symmetries of a regular toroidal map was accomplished. In this paper we complete this investigation on a surface of genus 1 considering the group of a regular toroidal hypermap of type $(3,3,3)$ that is a subgroup of index $2$ of the group of symmetries of a toroidal map of type $\{6,3\}$.