Uniform colorings and symmetry

The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123.

The cube has three classes of symmetry, which can be represented by vertex-transitive coloring the faces. The highest octahedral symmetry O_h has all the faces the same color. The dihedral symmetry D_4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D_2h is also a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol.

Name	Regular hexahedron	Square prism	Cuboid	Trigonal trapezohedron
Coxeter-Dynkin
Schläfli symbol	{4,3}	{4}x{}	{}x{}x{}
Wythoff symbol	3 | 4 2	4 2 | 2	| 2 2 2
Symmetry	Oh (*432)	D4h (*422)	D2h (*222)	D3d (2*3)
Symmetry order	24	16	8	12
Image (uniform coloring)	(111)	(112)	(123)	(111), (112), (122), and (222)

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