Binary Icosahedral Group

In mathematics, the binary icosahedral group 2I or <2,3,5> 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 a cyclic group of order 2, and is the preimage 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.)