Octahedral symmetry - Wikipedia

Abstraction: Symmetry group of the cube and octahedron with 48 total isometries

Key points:

• Regular octahedron (and cube) has 24 rotational symmetries and 48 total isometries (including reflections/inversions)
• Chiral octahedral group O (order 24) is isomorphic to S4 (symmetric group on 4 objects, corresponding to 4 body diagonals of cube)
• Full octahedral group O_h (order 48) is isomorphic to S4 × Z2; it is the hyperoctahedral group for n=3
• The 48 elements come from 3×3 permutation matrices with ±1 signs: 24 have det +1 (rotations), 24 have det −1 (reflections/inversions)
• Three reflectional generators from Coxeter-Dynkin diagram produce all rotational generators via products
• Achiral octahedral symmetry is among the crystallographic point groups of the cubic crystal system

Connections: Group Theory · Symmetry Groups

Source: http://en.wikipedia.org/wiki/Octahedral_symmetry