  All possible permutations in the discrete $S_4$ group are classified by three
rotation angles associated with the orthogonal group $O(3)$. We construct a
spinor representation ${\bf 2}_D$ of $O(3)$, which is transformed by three
4$\times$4 matrices corresponding to three Pauli matrices in $SO(3)$. An
irreducible decomposition of ${\bf 2}_D \otimes {\bf 2}_D$ supplies a vector
representation of {\bf 3} of $O(3)$, thereby, of $S_4$. Our construction is
consistent with the mathematical fact that $O(3)=SO(3)\times \boldsymbol{Z}_2$.
The $\boldsymbol{Z}_2$ parity in the spinorial space is described by a block
off-diagonal matrix as the spinorial parity operator, whose eigenvalues are
$\pm 1$ consistent with $\boldsymbol{Z}_2$.
