Group Structure
The icosahedral rotation group I is of order 60. The group I is isomorphic to A5, 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 Th with 20 versions of D3 (10 axes, 2 per axis), and 6 versions of D5.
The full icosahedral group Ih has order 120. It has I as normal subgroup of index 2. The group Ih is isomorphic to I × C2, or A5 × C2, with the inversion in the center corresponding to element (identity,-1), where C2 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).
Read more about this topic: Icosahedral Symmetry
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