Intransitive collineation groups of ovals fixing a triangle

We investigate collineation groups of a finite projective plane of odd order fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which there exists a fixed triangle off the oval is considered in detail. Our main result is the following. Theorem. Let π be a finite projective plane of odd order n containing an oval Ω. If a collineation group G of π satisfies the properties: (a) s Ω and the action of G on Ω yields precisely two orbits Ω1 and Ω2, even order and a faithful primitive action on Ω2, s neither points nor lines but fixes a triangle ABC in which the points A, B, C are not on the oval Ω, ∈{7,9,27}, the orbit Ω2 has length 4 and G acts naturally on Ω2 as A4 or S4. rder n∈{7,9,27} does furnish at least one example for the above situation; the determination of the planes and the groups which do occur is complete for n=7, 9; the determination of the planes is still incomplete for n=27.