What is the Weyl group of this group?

Question

Let $G$ be the group $GL_{n_1}(q^{l_1})\times GL_{n_2}(q^{l_2})\times GL_{n_3}(q^{l_3})$. Here $GL_{n_1}(q^{l_1})$ is the rational points of $GL(n,\bar {\mathbf{F}}_q)$. My question iis what is the Weyl group $W$ of $G$, and is there a reference to compute it. I believe that it is not just $S=\operatorname{Sym}_{n_1}\times \operatorname{Sym}_{n_2}\times\operatorname{Sym}_{n_3}$, there is some extra factors in $W$ other than $S$.