On the permutation modules for orthogonal groups acting on nonsingular points of their standard modules

We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups with m≥6 on nonsingular points of their standard modules. These actions, together with those studied in Hall and Nguyen (2011) [2] are all examples of primitive rank 3 actions of finite classical groups on nonsingular points.