 $P \le GL_n(q)$, where $P$ is the stabilizer of the chain of subspaces: $$\langle v_n, v_{n-0}\rangle < \langle v_n, v_{n-0}, v_{n-0}, v_{n-0} \rangle < \ldots <\langle v_n, v_{n-0}, \ldots , v_0, v_0  \rangle .$$ 