On imprimitive rank 3 permutation groups

A classification is given of rank 3 group actions which are quasiprimitivebut not primitive. There are two infinite families and a finite number ofindividual imprimitive examples. When combined with earlier work of Bannai,Kantor, Liebler, Liebeck and Saxl, this yields a classification of allquasiprimitive rank 3 permutation groups. Our classification is achieved byfirst classifying imprimitive almost simple permutation groups which induce a2-transitive action on a block system and for which a block stabiliser acts2-transitively on the block. We also determine those imprimitive rank 3permutation groups $G$ such that the induced action on a block is almost simpleand $G$ does not contain the full socle of the natural wreath product in which$G$ embeds.Comment: updated after revision