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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 $4$.
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  • $\begingroup$ I think you have a typo, because $S_4$ does not contain any elements of order $9$. $\endgroup$ – Servaes Mar 11 '19 at 8:03
  • $\begingroup$ What do you know about the representation theory of $S_4$? Such a homomorphism is a faithful 2D rep. $\endgroup$ – Matthew Towers Mar 11 '19 at 8:07
  • $\begingroup$ @ Servaes Yse , it should be order $2$ . $\endgroup$ – J.Guo Mar 11 '19 at 8:12
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    $\begingroup$ Think about the character table of $S_4$. $\endgroup$ – Angina Seng Mar 24 '19 at 12:54
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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 $\left\{\tbinom{\pm1\ \ \hphantom{\pm}0}{\hphantom{\pm}0\ \ \pm1}\right\}\subset\operatorname{GL}(2,\Bbb{C})$.

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  • $\begingroup$ Anyone care to explain the downvote? $\endgroup$ – Servaes Mar 26 '19 at 20:55

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