= Tetrahedron stabilizer in the cube rotation group
{title2=$|G_T|=12$}
For cube vertices $(\pm1,\pm1,\pm1)$, the four vertices of coordinate product $+1$ and the four of product $-1$ form the two inscribed <regular tetrahedra>. A cube <rotation> is a <signed permutation matrix> of <determinant> one and maps each product class to one class. A quarter-turn about a coordinate axis exchanges the classes. The <orbit> of either tetrahedron therefore has size two, while the <cube rotation group> has order 24. The <orbit-stabilizer theorem> gives its setwise <stabilizer subgroup> order 12; this is the tetrahedron's proper <rotation> <group>.
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