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

Question

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 $4$.

Answer 1

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 $\left\{\tbinom{\pm1\ \ \hphantom{\pm}0}{\hphantom{\pm}0\ \ \pm1}\right\}\subset\operatorname{GL}(2,\Bbb{C})$.