  We show that for all integers $m\geq 2$, and all integers $k\geq 2$, the
orthogonal groups $\Orth^{\pm}(2m,\Fk)$ act on abstract regular polytopes of
rank $2m$, and the symplectic groups $\Sp(2m,\Fk)$ act on abstract regular
polytopes of rank $2m+1$.
