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Concept
Binary icosahedral group
Summary
In mathematics, the binary icosahedral group 2I or is a certain nonabelian group of order 120.
It is an extension of the icosahedral group I or (2,3,5) of order 60 by the cyclic group of order 2, and is the of the icosahedral group under the 2:1 covering homomorphism
of the special orthogonal group by the spin group. It follows that the binary icosahedral group is a discrete subgroup of Spin(3) of order 120.
It should not be confused with the full icosahedral group, which is a different group of order 120, and is rather a subgroup of the orthogonal group O(3).
The binary icosahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Explicitly, the binary icosahedral group is given as the union of all even permutations of the following vectors:
8 even permutations of
16 even permutations of
96 even permutations of
Here 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(1).
The 120 elements in 4-dimensional space match the 120 vertices the 600-cell, a regular 4-polytope.
The binary icosahedral group, denoted by 2I, is the universal perfect central extension of the icosahedral group, and thus is quasisimple: it is a perfect central 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 { ±1 } by I. In fact, there is no subgroup of 2I isomorphic to I.
The center of 2I is the subgroup { ±1 }, so that the inner automorphism group is isomorphic to I. The full automorphism group is isomorphic to S5 (the symmetric group on 5 letters), 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 (different lifts give the same automorphism).
Official source
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