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

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.

• Why those specific groups?
– anon
Mar 19 at 21:47

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 $${\rm PSL}(2,7)$$.

• Thank you very much for your brilliant answer. Can you please explain me how you have constructed this map?
– PAMG
Mar 19 at 17:02
• It was a computer calculation, which you can do in GAP or Magma. I just computed the irreducible representations of ${\rm SL}(2,7)$ over ${\mathbb F}_{11}$, which is an easy calculation, and then wrote down two generating matrices from the image of a $4$-dimensional irreducible. Mar 19 at 19:06
• The GAP command for this is IrreducibleRepresentations(G,GF(11)). Mar 19 at 20:24
• @DerekHolt Thank you very much. You are amazing.
– PAMG
Mar 20 at 1:46
• @ahulpke Thank you for sharing this command.
– PAMG
Mar 20 at 1:46