Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 8D / b / iii / Solution

There are four main diagonals and the action is transitive, so the stabilizer of one has orderFor the diagonal spanned by $(1,1,1)$, the stabilizing signed permutation matrices are $P$ and $-P$ with $P$ a permutation matrix. HenceThe action is not faithful: both $I$ and the central inversion $-I$ fix every main diagonal setwise. In fact its kernel is exactly $\{I,-I\}$.