Explicitly, the binary icosahedral group is given as the union of the 24 Hurwitz units with all 96 quaternions obtained from by an even permutation of all the four coordinates 0, 1, φ−1, φ, and with all possible sign combinations. Here φ = /2 is the golden ratio. In total there are 120 elements, namely the unit icosians. They all have unit magnitude and therefore lie in the unit quaternion group Sp. The 120 elements in 4-dimensional space match the 120 vertices the 600-cell, a regular 4-polytope.
Properties
Central extension
The binary icosahedral group, denoted by 2I, is the universal perfect central extension of the icosahedral group, and thus is quasisimple: it is a perfectcentral extension of a simple group. Explicitly, it fits into the short exact sequence This sequence does not split, meaning that 2I is not a semidirect product of by I. In fact, there is no subgroup of 2I isomorphic to I. The center of 2I is the subgroup, so that the inner automorphism group is isomorphic to I. The full automorphism group is isomorphic to S5, just as for - any automorphism of 2I fixes the non-trivial element of the center, hence descends to an automorphism of I, and conversely, any automorphism of I lifts to an automorphism of 2I, since the lift of generators of I are generators of 2I.
Superperfect
The binary icosahedral group is perfect, meaning that it is equal to its commutator subgroup. In fact, 2I is the unique perfect group of order 120. It follows that 2I is not solvable. Further, the binary icosahedral group is superperfect, meaning abstractly that its first two group homology groups vanish: Concretely, this means that its abelianization is trivial and that its Schur multiplier is trivial. In fact, the binary icosahedral group is the smallest superperfect group. The binary icosahedral group is not acyclic, however, as Hn is cyclic of order 120 for n = 4k+3, and trivial for n > 0 otherwise,.
Isomorphisms
Concretely, the binary icosahedral group is a subgroup of Spin, and covers the icosahedral group, which is a subgroup of SO. Abstractly, the icosahedral group is isomorphic to the symmetries of the 4-simplex, which is a subgroup of SO, and the binary icosahedral group is isomorphic to the double cover of this in Spin. Note that the symmetric groupdoes have a 4-dimensional representation, and that the full symmetries of the 4-simplex are thus not the full icosahedral group. The binary icosahedral group can be considered as the double cover of the alternating group denoted this isomorphism covers the isomorphism of the icosahedral group with the alternating group. Just as is a discrete subgroup of, is a discrete subgroup of the double over of, namely. The 2-1 homomorphism from to then restricts to the 2-1 homomorphism from to. Similarly, is a discrete subgroup of, and its two double covers are discrete subgroups of the two Pin groups. One can show that the binary icosahedral group is isomorphic to the special linear group SL — the group of all 2×2 matrices over the finite fieldF5 with unit determinant; this covers the exceptional isomorphism of with the projective special linear group PSL. Note also the exceptional isomorphism which is a different group of order 120, with the commutative square of SL, GL, PSL, PGL being isomorphic to a commutative square of which are isomorphic to subgroups of the commutative square of Spin, Pin, SO, O.
Presentation
The group 2I has a presentation given by or equivalently, Generators with these relations are given by
Subgroups
The only proper normal subgroup of 2I is the center. By the third isomorphism theorem, there is a Galois connection between subgroups of 2I and subgroups of I, where the closure operator on subgroups of 2I is multiplication by. is the only element of order 2, hence it is contained in all subgroups of even order: thus every subgroup of 2I is either of odd order or is the preimage of a subgroup of I. Besides the cyclic groups generated by the various elements, the only other subgroups of 2I are:
binary dihedral groups, Dic5=Q20=⟨2,2,5⟩, order 20 and Dic3=Q12=⟨2,2,3⟩ of order 12
The quaternion group, Q8=⟨2,2,2⟩, consisting of the 8 Lipschitz units forms a subgroup of index 15, which is also the dicyclic group Dic2; this covers the stabilizer of an edge.
The 4-dimensional analog of the icosahedral symmetry groupIh is the symmetry group of the 600-cell. Just as the former is the Coxeter group of type H3, the latter is the Coxeter group of type H4, also denoted . Its rotational subgroup, denoted Coxeter notation#Rank four groups|+ is a group of order 7200 living in SO. SO has a double cover called Spin in much the same way that Spin is the double cover of SO. Similar to the isomorphism Spin = Sp, the group Spin is isomorphic to Sp × Sp. The preimage of + in Spin is precisely the product group 2I × 2I of order 14400. The rotational symmetry group of the 600-cell is then Various other 4-dimensional symmetry groups can be constructed from 2I. For details, see.
