= Rotational symmetry group of a cube
{title2=$|H|=24$}
= Cube rotation group
{synonym}
The orientation-preserving symmetries of a cube are the $3\times3$ <signed permutation matrices> with <determinant> one. There are $3!\,2^2=24$ of them. Their <group action> on the four body diagonals identifies this <group> with the <symmetric group> $S_4$. Indeed choose diagonal directions $(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)$, whose sum is zero and whose only linear relation has all coefficients equal. A rotation fixing every diagonal sends each direction to itself or its negative; the relation forces all four signs equal. All negative would give $-I$, which has <determinant> $-1$, so the <kernel of a group homomorphism> is trivial. Both <groups> have order 24, giving the identification. Its face <permutations> have cycle types $1^6$, $1^24$, $1^22^2$, $2^3$ and $3^2$, occurring $1,6,3,6,8$ times respectively.
Back to article page
