A symmetry cube is a fascinating concept in geometry. A cube has 12 edges, and any symmetry of the cube permutes these edges. The cube has four classes of symmetry, represented by vertex-transitive coloring the faces. The highest octahedral symmetry has all the faces the same color. The dihedral symmetry comes from the cube being a solid, with all the six sides being different colors. The group of symmetries of a cube is isomorphic to S4 Z=2, representing the permutations of the four diagonals of the cube.