= Cube symmetry action on edges
{title2=$H\hookrightarrow S_{12},\quad |H|=48$}
The full <symmetry group of a cube> acts transitively and faithfully on its twelve edges. For vertices $(\pm1,\pm1,\pm1)$, its <signed permutation matrices> act transitively on the edges. The <stabilizer subgroup> of $\{(1,1,z):-1\le z\le1\}$ consists of independently swapping $x,y$ and reversing $z$, and is a <Klein four-group>. The <orbit-stabilizer theorem> gives order $12\cdot4=48$. Fixing every edge fixes every vertex, as each vertex is the unique intersection of its three incident edges, so the <group action> is faithful. The resulting <subgroup> of $S_{12}$ has index $12!/48$ and is not normal: central inversion has <cycle type> $2^6$, whose <conjugacy class> in $S_{12}$ has $12!/(2^6 6!)=10395>48$ elements.
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