Let $ G $ be a finite subgroup of $ SU(n) $. Suppose that $ G $ is not contained in any positive dimensional subgroup of $ SU(n) $. Then certainly $ G $ is irreducible. Must it also be the case that the derived subgroup $ G'=[G,G] $ is irreducible?
More generally, let $ G $ a finite primitive subgroup of $ SU(n) $. Is $ G' $ always irreducible?
Recall that we say a subgroup $ G $ of $ GL_n(\mathbb{C}) $ is imprimitive if we can write $ \mathbb{C}^n=V_1 \oplus \dots \oplus V_k $ as a direct sum of smaller subspaces such that every $ g \in G $ just permutes the subspaces. In other words, for any $ g \in G $ the subspaces $ g(V_1) \oplus \dots \oplus g(V_k) $ are just a permutation of $ V_1 \dots V_k $. That is, $ g(V_i)= V_{\sigma(i)} $. If no such decomposition is possible then we say that $ G $ is primitive.
Both claims are true for $ SU(2) $. Both claims also appear to be true for examples I can think of for $ SU(3) $ and $ SU(4) $.