= Solution
Use the <group representation> generated by the displayed matrices. The coordinate reflections generate $(\mathbb Z_2)^3$, and the cyclic permutation conjugates $\kappa_x$ to $\kappa_y$, then to $\kappa_z$. Thus the group is the <semidirect product> $(\mathbb Z_2)^3\rtimes\mathbb Z_3$, of order $24$. The converted TeX's multiplication sign must not be interpreted as a direct product of commuting factors.
To prove absolute irreducibility, complexify to $\mathbb C^3$ and let $W$ be a nonzero complex <invariant subspace>. It is preserved by the coordinate projections $P_j=(I-\kappa_j)/2$. For a nonzero $w\in W$, at least one coordinate is nonzero, so $P_jw$ is a nonzero multiple of a coordinate vector. Applying the cyclic permutation repeatedly shows that all three coordinate vectors belong to $W$. Hence $W=\mathbb C^3$. Therefore \b[the real representation is absolutely irreducible].
Equivalently, a commuting matrix must be diagonal because it commutes with all three reflections; commuting with $\rho$ forces its diagonal entries equal. This also confirms that every equivariant linearization is a scalar multiple of the identity, the condition needed in the branching calculation.
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