Rotational symmetry | Finite groups

Octahedral symmetry

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual to an octahedron. The group of orientation-preserving symmetries is S4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube. (Wikipedia).

Octahedral symmetry
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Hexagon | Icosahedral symmetry | Rhombicuboctahedron | Translational symmetry | If and only if | Bolza surface | Truncated cuboctahedron | Hyperoctahedral group | Platonic solid | Cuboid | Orthographic projection | Tetrahedral symmetry | Index of a subgroup | Tetrakis hexahedron | Systolic geometry | Disdyakis dodecahedron | Wreath product | Snub cube | Archimedean solid | Improper rotation | Symmetric group | Truncated octahedron | Alternating group | Determinant | Pentagonal icositetrahedron | Harold Scott MacDonald Coxeter | Symmetry number | Orbifold notation | Direct product of groups | Compound of cube and octahedron | List of small groups | Riemann surface | Symmetry group | Square | Dihedral group | Point groups in three dimensions | Prism (geometry) | Rhombic dodecahedron | Dihedral symmetry in three dimensions | Cube | Truncated cube | Quaternions and spatial rotation | Cyclic group | Cuboctahedron | Stereographic projection | Binary octahedral group | Dice | Coxeter notation | Triakis octahedron | Cubic crystal system | Octahedron | Deltoidal icositetrahedron | Permutation matrix | Dual polyhedron | Conjugacy class | Schoenflies notation | Crystal system | Catalan solid