Character of 2-dimensional irreducible representation of $S_4$

Consider the 2-dimensional irreducible representation of $S_4$, obtained by composing the standard 2-dimensional irreducible representation of $S_3$ with the surjective homomorphism $f:S_4 \rightarrow S_3, (1  2) \mapsto (1  2), (2  3) \mapsto (1  3), (3   4) \mapsto (1  2)$. Please compute its character directly, without using character table orthogonality relations. Thanks!

  • This is not my area of expertise, but your deadline seems too short for the level of your question and the offered bounty.

  • Your text has typos, it’s not clear what you mean. You want the 3d rep. of. S3 Not S4. The result is then not a 2d rep but a 3d rep, is this what you mean?

  • @Dynkin, sorry yes, I edited the text

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  • Sorry, it is actually May 20, 10 a.m. GMT +1. Please let me know if you see this and are interested.

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The answer is accepted.