= Tetrahedral symmetry
{title2=$T\cong A_4,\quad T_d\cong S_4$}
A regular tetrahedron has a proper rotational symmetry group $T$ of order 12, isomorphic to the <alternating group> $A_4$ acting on its four vertices. Its full spatial symmetry group $T_d$ has order 24 and is isomorphic to the <symmetric group> $S_4$; the additional operations reverse orientation. Three-fold vertex/face-axis rotations and half-turns about opposite-edge midpoints generate the proper group. The distinction matters when a field symmetry uses only proper rotations, but a scalar density also admits reflections.
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