Rank 3 Permutation Characters and Maximal Subgroups

Let G be a transitive permutation group acting on a finite set E and let P be the stabilizer in G of a point in E. We say that G is primitive rank 3 on E if P is maximal in G and P has exactly three orbits on E. For any subgroup H of G, we denote by 1H the permutation character (or permutation module) over C of G on the cosets G/H. Let H and K be subgroups of G. We say 1H 6 1 G K if 1 G K − 1 G H is a character of G. Also a finite group G is called nearly simple primitive rank 3 on E if there exists a quasi-simple group L such that L/Z(L)EG/Z(L) 6 Aut(L/Z(L)) and G acts as a primitive rank 3 permutation group on the cosets of some subgroup of L. In this paper we classify all maximal subgroups M of a nearly simple primitive rank 3 group G of type L = Ω2m+1(3),m > 3, acting on an L-orbit E of non-singular points of the natural module for L such that 1P 6 1 G M , where P is the stabilizer of a non-singular point in E. This result has an application to the study of minimal genera of algebraic curves which admit group actions.