Rank three innately transitive permutation groups and related $2$-transitive groups

Abstract

The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of finite transitive permutation groups, and are analogues of composition factors for abstract finite groups. This paper extends classifications of finite primitive and quasiprimitive groups of rank at most $3$ to a classification for the finite innately transitive groups. The new examples comprise three infinite families and three sporadic examples. A necessary step in this classification was the determination of certain configurations in finite almost simple $2$-transitive groups called special pairs.