The structure of rank 3 permutation modules for O_{2n}(2) and U_m(2) acting on nonsingular points

We study the odd-characteristic structure of permutation modules for the rank 3 natural actions of $O_{2n}^{\pm }(2)$ ($n\ge 3$) and $U_m(2)$ ($m\ge 4$) on nonsingular points of their standard modules.