Quiz:Linear representation theory of alternating group:A4

For background, see linear representation theory of alternating group:A4.

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Let $\omega$ be a primitive cube root of unity. The character table over a splitting field is as follows:

Representation/conjugacy class representative	$()$ (size 1)	$(1,2)(3,4)$ (size 3)	$(1,2,3)$ (size 4)	$(1,3,2)$ (size 4)
trivial representation	1	1	1	1
first nontrivial one-dimensional representation	1	1	$\omega$	$\omega ^{2}$
second nontrivial one-dimensional representation	1	1	$\omega ^{2}$	$\omega$
three-dimensional irreducible representation	3	-1	0	0

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