Several books on Group Theory show that the isomorphism between the group of proper rotations of a cube and P4, the group of 24 permutations on four arbitrary symbols, can be established by a consideration of the four diagonals of the cube. To each arrangement of these diagonals corresponds a unique position of the cube and therefore a unique rotation from one position as reference to that position. The extension of this result to other polyhedra does not appear in the books that I know.
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