= Induction of the planar square representation to the symmetric group on four letters
{title2=$\operatorname{Ind}_{D_8}^{S_4}W\cong C\oplus(C\otimes\operatorname{sgn})$}
Embed the order-eight <dihedral group> as $H=\langle(1234),(12)(34)\rangle$ in the <symmetric group> $S_4$, and let $W$ be its two-dimensional <complex representation> obtained from its <square> action. Let $C$ be the three-dimensional <complex representation> from the <rotational symmetry group of a cube>. In the class order $1,(12),(12)(34),(123),(1234)$, the <induced character> is $(6,0,-2,0,0)$: only the identity and the square's half-turn have nonzero <character> values. This equals the <character> of $C\otimes(\mathbf1\oplus\operatorname{sgn})$, since $\chi_C=(3,-1,-1,0,1)$. <Characters distinguish finite-dimensional complex semisimple representations>, giving the displayed <isomorphism>. Its <trivial character> multiplicity is $(6-6)/24=0$.
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