Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 60 / 1 / c / Solution

Conjugating the coordinate reflection by the rotations supplies reflections in the other two coordinates. The quarter-turns, together with those reflections, also supply coordinate transpositions. Consequently the represented group consists of all signed permutation matrices, the full symmetry group of a cube of order $2^3\cdot 3!=48$.

Let $C$ be any real or complex matrix commuting with the group representation. Commutation with the three independent coordinate reflections forces all off-diagonal entries of $C$ to vanish. Commutation with the quarter-turn about the first axis equates its second and third diagonal entries, and the quarter-turn about the second axis equates its first and third entries. ThereforeFor completeness, if the complexified group representation had a proper invariant subspace, its orthogonal complement would be invariant too, since the generating matrices are unitary. The orthogonal projection onto that subspace would be a nonscalar commuting matrix, contradicting the calculation. Thus the complexification is irreducible, proving absolute irreducibility of the real group representation.