  A classification is given of rank 3 group actions which are quasiprimitive
but not primitive. There are two infinite families and a finite number of
individual imprimitive examples. When combined with earlier work of Bannai,
Kantor, Liebler, Liebeck and Saxl, this yields a classification of all
quasiprimitive rank 3 permutation groups. Our classification is achieved by
first classifying imprimitive almost simple permutation groups which induce a
2-transitive action on a block system and for which a block stabiliser acts
2-transitively on the block. We also determine those imprimitive rank 3
permutation groups $G$ such that the induced action on a block is almost simple
and $G$ does not contain the full socle of the natural wreath product in which
$G$ embeds.

Group Theory