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So, as per one of my previous questions, I'm working through some problems in Armstrong's book Groups and Symmetry. The first two thirds of the question I've managed to grind through (after much spatially-related struggle) but here I seem to be--quite literally--drawing a blank. Can anyone guide me toward a geometric object whose (full) symmetry group corresponds to $A_4\times\mathbb Z_2$? I know that $A_4$ corresponds to the proper rotations of a tetrahedron, but its full symmetry group $S_4$ isn't isomorphic to the product group in question here. Any help you can offer me is much appreciated!

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(Pictures from Wikipedia.)

An object in three dimensions with symmetry group isomorphic to $A_4\times\mathbb{Z}_2$ is said to have pyritohedral symmetry. This is the symmetry of a volleyball:

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Equivalently, it is the symmetry of a cube whose faces have been subdivided in a certain way:

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Polyhedra with this symmetry group include the pyritohedron, which is a certain type of irregular dodecahedron:

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Pyritohedral symmetry is so-named because it is the point symmetry group for one of the crystal structures of the mineral pyrite.

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    @AsinglePANCAKE: No, not as far as I know. But "van Leeuwen" is a very common name in some places.2011-12-05