= Body-diagonal stabilizer in the cube symmetry group
{title2=$H\cong S_3\times C_2$}
For vertices $(\pm1,\pm1,\pm1)$, the <symmetry group of a cube> consists of <signed permutation matrices>. A matrix preserving the unoriented line through $(1,1,1)$ must have all three signs equal, so it has the form $\varepsilon P$, where $P$ is a <permutation matrix> and $\varepsilon\in\{1,-1\}$. Central inversion commutes with every $P$ and is not itself a permutation matrix. Consequently $(P,\varepsilon)\mapsto\varepsilon P$ identifies this twelve-element <stabilizer subgroup> with the <direct product of groups> $S_3\times C_2$. A body diagonal is different from a face-normal coordinate axis, whose line stabilizer has order sixteen.
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