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 the cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism

${\displaystyle \operatorname {Spin} (3)\to \operatorname {SO} (3)\,}$

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).

In the algebra of quaternions, the binary icosahedral group is concretely realized as a discrete subgroup of the versors, which are the quaternions of norm one. For more information see Quaternions and spatial rotations.