A non-trivial homomorphism from $SL(2, 7)$ to $GL(4, 11)$.

Question

Let $G=GL(4, 11)$ be the general linear group of $4\times 4$ matrices over the field $\mathbb{F}_{11}$. Let $H=SL(2, 7)$ be a special linear group of $2\times 2$ matrices over the field $\mathbb{F}_{7}$. I want to know whether there is any non-trivial group homomorphism $\phi$, where $$\phi:H\to G.$$ I know that order of $SL(2, 7)$ divides order of $G$ (or $|G|$). Therefore, such a non-trivial homomorphism may exist as $\phi(H)$ divides $|H|$ as well as $|G|$. Next, I know that any homomorphism from $H$ can be defined only from its generators. But I am not being able to find it. Please help.

Answer 1

Yes, there is an injective homomorphism with image generated by the matrices $$ \left(\begin{array}{cccc}10&6&0&0\\0&1&1&0\\0&10&0&0\\0&4&0&10\end{array}\right),\quad \left(\begin{array}{cccc}0&1&1&0\\0&0&4&1\\0&0&1&0\\1&0&6&0\end{array}\right).$$ There is even a nontrivial homomorphism ${\rm SL}(2,7) \to {\rm GL}(3,11)$ but that has image isomorphic to ${\rm PSL}(2,7)$.