Character of 2dimensional irreducible representation of $S_4$
Consider the 2dimensional irreducible representation of $S_4$, obtained by composing the standard 2dimensional 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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Hey there, I hope you're alright! I was wondering if you could help me with some questions this Friday, 10a.m. GMT. I will have around a 22.5 hour window and I'm willing to pay $60 per question if you're online and able to answer within the time limit, would you be interested?

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

Sure, I‘ll keep an eye out, but I can‘t guarantee that I’ll have time.
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