Note that other presentations are possible, for instance as an alternating group (for "I"). The ' I""' is of order 60. The group "I" is isomorphic to "A"5, the alternating group of even permutations of five objects. This isomorphism can be realized by "I" acting on various compounds, notably the compound of five cubes (which inscribe in the dodecahedron), the compound of five octahedra, or either of the two compounds of five tetrahedra (which are enantiomorphs, and inscribe in the dodecahedron). The group contains 5 versions of "T"h with 20 versions of "D3" (10 axes, 2 per axis), and 6 versions of "D5". The ' Ih""' has order 120. It has "I" as normal subgroup of index 2. The group "Ih" is isomorphic to "I" × "Z"2, or "A"5 × "Z"2, with the inversion in the center corresponding to element (identity,-1), where "Z"2 is written multiplicatively. "Ih" acts on the compound of five cubes and the compound of five octahedra, but −1 acts as the identity (as cubes and octahedra are centrally symmetric). It acts on the compound of ten tetrahedra: "I" acts on the two chiral halves (compounds of five tetrahedra), and −1 interchanges the two halves. Notably, it does "not" act as S5, and these groups are not isomorphic; see below for details. The group contains 10 versions of "D3d" and 6 versions of "D5d" (symmetries like antiprisms). "I" is also isomorphic to PSL2(5), but "Ih" is not isomorphic to SL2(5). The following groups all have order 120, but are not isomorphic: They correspond to the following short exact sequences (the latter of which does not split) and product In words, Note that formula_6 has an exceptional irreducible 3-dimensional representation (as the icosahedral rotation group), but formula_7 does not have an irreducible 3-dimensional representation, corresponding to the full icosahedral group not being the symmetric group. These can also be related to linear groups over the finite field with five elements, which exhibit the subgroups and covering groups directly; none of these are the full icosahedral group:  All of these classes of subgroups are conjugate (i.e., all vertex stabilizers are conjugate), and admit geometric interpretations.