The properties of the group are contained in its product table. In general, it is rather tedious to deal with the full group table. Each row of the table provides a permutation of all 120 elements. Geometrically, this corresponds to a rotation or reflexion of the polyhedron.

Let us hav a closer look at the expanded truncated icosahedron. It consists of 120 vertices grouped in squares, hexagons and decagons.

There are three ways to shrink it to a polyhedron with fewer vertices, by identifying two neighbouring vertices:

1. Shrink the square to a line and the hexagons to triangles. One ends up with a truncated dodecahedron as illustrated in this animation.
2. Shrink the square to a line in the other direction (the hexagons stay and the decagons turn into pentagons). One ends up with a truncated icosahedron (soccer ball).
3. Keep the squares and shrink hexagons to triangles and the decagons to pentagons. This generates the edge-truncated icosahedron as illustrated in this animation.

In each case, the number of vertices is reduced to 60. 

Similarly, the truncated dodecahedron (60 vertices) can shrink to the icosidodecahedron (30 vertices). The rhombicosidodecahedron (60 vertices) can shrink to the dodecahedron (20 vertices) or to the icosahedron (12 vertices).