Rot(Cube) ≅ S₄ via action on body diagonals

Each rotation permutes the four body diagonals, giving a homomorphism φ: G → S₄. If all diagonals are fixed as sets, every vertex is fixed ⇒ identity; |G|=|S₄|=24 ⇒ isomorphism.

Bottom: An embedding of the Cayley graph of S₄ with generators s₁=(12), s₂=(23), s₃=(34).

diag 1 diag 2 diag 3 diag 4
Induced permutation on diagonals: [1,2,3,4]
σ (cycle form): e
two-row:
[ 1 2 3 4 ]
[ 1 2 3 4 ]