One can inscribe 5 tetrahedra within a dodecahedron. In this VRML model the five tetrahedra are colored yellow, green, blue, violet and red. A symmetry of the dodecahedron can be reinterpreted as a permuation of these five colors.
It is fairly easy to count that the dodecahedron has a symmetry group of order 60. (It can be placed on each of its 12 faces in 5 orientations.) It is also well known that the Alternating group of degree n (the even permutations of n objects) has n!/2 elements. Thus it is very plausible that the symmetry group of the dodecahedron is isomorphic to A5 since 5!/2 = 60.
The even permutations of 5 things are either odd cycles (in particular 3-cycles and 5-cycles) or pairs of disjoint transpositions. Links to VRML models illustrating symmetries having these cycle structures are below.
Help on viewing VRML models can be obtained from the NIST.