Title: The Modulo 2 Structure of rank 3 permutation modules for 
 odd characteristic symplectic groups
 Authors: Jeffrey Lataille, Peter Sin and Pham Huu Tiep
                        Department of Mathematics
                        University of Florida 
                        358 Little Hall
                        PO Box 118105
                        Gainesville, FL 32611-8105 

Abstract.  

This paper studies the permutation representation of the symplectic group 
${\rm Sp}(2{\rm m},\mathbb F_q),$ 
where $q$ is odd, on the 1-spaces of its natural module.  
The complete submodule lattice for the modulo $\ell$ 
reduction of this permutation module is known for all odd primes $\ell$ 
not dividing $q.$
In this paper we determine the complete submodule lattice 
for the mod 2 reduction. Similar results are then obtained for the orthogonal group 
${\rm O}(5, \mathbb F_q).$ 

Preprint