The non-synchronizing primitive groups of degree up to 624

TL;DR

This paper classifies non-synchronizing primitive permutation groups of degree up to 624, using theoretical and computational methods to identify non-separating groups and discover new examples of non-synchronizing structures.

Contribution

It provides a comprehensive classification of non-synchronizing primitive groups up to degree 624 and introduces new examples and sequences of such groups.

Findings

Identified all non-synchronizing primitive groups up to degree 624.

Discovered a new infinite sequence of non-synchronizing graphs.

Found examples of non-separating but synchronizing groups.

Abstract

We describe the methods and results of a classification of the non-synchronizing primitive permutation groups of degree up to 624. We make use of theory and computation to determine the primitive groups of degree up to 624 that are non-separating, which is a necessary, and very often sufficient, condition for a primitive group to be non-synchronizing, and determine exactly which of these non-separating groups are non-synchronizing. A new infinite sequence of non-synchronizing graphs is discovered and new examples of non-separating, but synchronizing, groups are found.