Finite 2-groups having a cyclic or dihedral maximal subgroup and arc-transitive maps

TL;DR

This paper classifies finite 2-groups with specific maximal subgroups and explores their automorphisms to classify certain arc-transitive maps with specific Euler characteristic properties.

Contribution

It provides a complete classification of finite 2-groups with cyclic or dihedral maximal subgroups and characterizes associated arc-transitive maps with particular Euler characteristics.

Findings

Classified all finite 2-groups with cyclic or dihedral maximal subgroups.

Determined automorphism groups of these 2-groups.

Classified all G-arc-transitive maps with Euler characteristic not divisible by 4.

Abstract

We classify all finite 2-groups that have a cyclic or dihedral maximal subgroup and determine their automorphism groups. Based on this result, we classify all pairs $ (G,\mathcal{M}) $, such that $ G $ is a finite 2-group and $ \mathcal{M} $ is a $ G $-arc-transitive map with Euler characteristic not being divisible by 4.