# Is there an injective homomorphism from $S_4$ to $GL(2,C)$

Is there an injective homomorphism from $$S_4$$ to $$GL(2,C)$$?

My attempt :

1. If such an injective homomorphism exists, then $$S_4$$ is isomorphic to a subgroup $$A$$ of $$GL(2,C)$$.
2. $$A$$ must contain nine elements of order $$2$$; eight elements of order $$3$$ and six elements of order $$4$$.
• I think you have a typo, because $S_4$ does not contain any elements of order $9$. – Inactive - avoiding CoC Mar 11 at 8:03
• What do you know about the representation theory of $S_4$? Such a homomorphism is a faithful 2D rep. – Matthew Towers Mar 11 at 8:07
• @ Servaes Yse , it should be order $2$ . – J.Guo Mar 11 at 8:12
• Think about the character table of $S_4$. – Lord Shark the Unknown Mar 24 at 12:54

Hint: Because $$S_4$$ has a normal subgroup isomorphic to the Klein four group, after a change of basis the image of $$S_4$$ is contained in the normalizer of the subgroup $$\left\{\tbinom{\pm1\ \ \hphantom{\pm}0}{\hphantom{\pm}0\ \ \pm1}\right\}\subset\operatorname{GL}(2,\Bbb{C})$$.