The symmetry group here is

The 24-cell has 24 vertices, 96 edges, 96 triangles and 24 octahedra.

**The 600-cell**
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The symmetry group here is **W(H4)**, and the polytope may
be constructed as the convex hull of the **H4** root system.
It has 120 vertices, 720 edges, 1200 triangles, and 600 tetrahedra.

**The 120-cell**
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This is the dual of the 600-cell; it has 600 vertices, 1200 edges,
720 pentagons, and 120 dodecahedra.

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