A remark on the permutation representations afforded by the embeddings of O_(2m)^pm(2^f) in Sp_(2m)(2^f)

We show that the permutation module over $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on its natural module is isomorphic to the permutation module over $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on the union of the right cosets of $O_{2m}^+(2^f)$ and $O_{2m}^-(2^f)$.