The transitive groups of degree 48 and some applications

TL;DR

This paper enumerates all transitive subgroups of degree 48, analyzes their properties, and applies these findings to classify vertex-transitive graphs and improve bounds on generating elements.

Contribution

It provides the first complete enumeration of transitive subgroups of degree 48 and applies this to graph classification and group generation bounds.

Findings

Enumerated 195,826,352 conjugacy classes of transitive subgroups of S48.

Identified 25,707 minimal transitive and 713 elusive groups.

Improved bounds on the number of elements needed to generate transitive groups.

Abstract

The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the $195\,826\,352$ conjugacy classes of transitive subgroups of the symmetric group $S_{48}$ of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive graphs of degree $48$, of which there are $1\,538\,868\,366$, all but $0.1625\%$ of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published by the third author as a separate paper