contains 15 elements, while the icosidodecahedron has 30 vertices.

The two conjugacy classes of twelve 5-cycles in A₅ are represented by two icosahedra, of radii 2/5 and 4/5, respectively. The nontrivial outer automorphism in Out(A₅) ? Z₂ interchanges these two classes and the corresponding icosahedra.

The 15 puzzle

It can be proved that the 15 puzzle, a famous example of the sliding puzzle,

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