Finite semiprimitive permutation groups of rank $3$

TL;DR

This paper provides a complete classification of finite semiprimitive permutation groups of rank 3 that are not innately transitive, expanding understanding beyond previously classified classes.

Contribution

It introduces a full classification of non-innately transitive semiprimitive groups of rank 3, including specific examples and exceptional cases.

Findings

Classified all finite semiprimitive groups of rank 3 not innately transitive.

Identified examples as Schur coverings of almost simple 2-transitive groups.

Discovered three exceptional small groups.

Abstract

A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either semiregular or transitive.The class of semiprimitive groups properly contains primitive groups, quasiprimitive groups and innately transitive groups.The latter three classes of groups of rank $3$ have been classified, forming significant progresses on the long-standing problem of classifying permutation groups of rank $3$.In this paper, a complete classification is given of finite semiprimitive groups of rank $3$ that are not innately transitive, examples of which are certain Schur coverings of certain almost simple $2$-transitive groups, and three exceptional small groups.