Let $A_4$ be the alternating group and let $V = {\{(1), (12)(34), (13)(24), (14)(23)}\}$ be a normal subgroup of $A_4$. Then $A_4/V \simeq C_3$, so $A_4$ has $3$ one dimensional representations, which apparently, we can inflate/lift from the character table for $C_3$.

I don't really understand how this works? How do I relate the character values of $C_3$ to $A_4$? Do I need to define a homomorphism from $A_4$ to $C_3$?

  • $\begingroup$ Just look at the representation $A_4 \to A_4 /V \to GL(X)$ induced from a representation $A_4/V \to GL(X)$. $\endgroup$ – anomaly Jan 21 at 22:00

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