Let $A_4$ act on the four long diagonals (labeled $1,2, 3, 4$) inscribed in a cube (which is $S_4$). Then $A_4$ acts on the faces, the edges, and vertices of the cube. This gives rise to three representations whose characters we denote by $\chi^{faces}$, $\chi^{edges}$, $\chi^{vertices}$.
(1) How would you express each of these characters as linear combinations of the irreducible characters (and representations) of $A_4$? (Give a visual explaination as well)
(2) $S_4$ acts by conjugation on the normal subgroup $A_4$. How does this action operate on the isomorphism classes of irreducible representations of $A_4$?