Solution
= Solution
There are four main diagonals and the action is transitive, so the stabilizer of one has order
$$
\frac{48}{4}=12.
$$
For the diagonal spanned by $(1,1,1)$, the stabilizing signed permutation <matrices> are $P$ and $-P$ with $P$ a permutation <matrix>. Hence
$$
\boxed{G_d\cong C_2\times S_3}.
$$
The action is not faithful: both $I$ and the central inversion $-I$ fix every main diagonal setwise. In fact its kernel is exactly $\{I,-I\}$.