On the doubly transitive permutation representations of )

Each symplectic group over the field of two elements has two exceptional doubly transitive actions on sets of quadratic forms on the defining symplectic vector space. This paper studies the associated 2-modular permutation modules. Filtrations of these modules are constructed which have subquotients which are modules for the symplectic group over an algebraically closed field of characteristic 2 and which, as such, have filtrations by Weyl modules and dual Weyl modules having fundamental highest weights. These Weyl modules have known submodule structures. It is further shown that the submodule structures of the Weyl modules are unchanged when restricted to the finite subgroups Sp(2n,2) and O±(2n,2).