Binary icosahedral group

It follows that the binary icosahedral group is a discrete subgroup of Spin(3) of order 120.

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.

Explicitly, the binary icosahedral group is given as the union of all even permutations of the following vectors: Here

In total there are 120 elements, namely the unit icosians.

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.

[2] 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.

The center of 2I is the subgroup { ±1 }, so that the inner automorphism group is isomorphic to I.

- any automorphism of 2I fixes the non-trivial element of the center (

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.

Concretely, this means that its abelianization is trivial (it has no non-trivial abelian quotients) and that its Schur multiplier is trivial (it has no non-trivial perfect central extensions).

[citation needed] The binary icosahedral group is not acyclic, however, as Hn(2I,Z) is cyclic of order 120 for n = 4k+3, and trivial for n > 0 otherwise, (Adem & Milgram 1994, p. 279).

Concretely, the binary icosahedral group is a subgroup of Spin(3), and covers the icosahedral group, which is a subgroup of SO(3).

Abstractly, the icosahedral group is isomorphic to the symmetries of the 4-simplex, which is a subgroup of SO(4), and the binary icosahedral group is isomorphic to the double cover of this in Spin(4).

[citation needed] The binary icosahedral group can be considered as the double cover of the alternating group

One can show that the binary icosahedral group is isomorphic to the special linear group SL(2,5) — the group of all 2×2 matrices over the finite field F5 with unit determinant; this covers the exceptional isomorphism of

with the projective special linear group PSL(2,5).

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(4), Pin(4), SO(4), O(4).

The group 2I has a presentation given by or equivalently, Generators in the group of unit quaternions with these relations are given by The only proper normal subgroup of 2I is the center { ±1 }.

Besides the cyclic groups generated by the various elements (which can have odd order), the only other subgroups of 2I (up to conjugation) are:[3] The 4-dimensional analog of the icosahedral symmetry group Ih is the symmetry group of the 600-cell (also that of its dual, the 120-cell).

Its rotational subgroup, denoted [3,3,5]+ is a group of order 7200 living in SO(4).

SO(4) has a double cover called Spin(4) in much the same way that Spin(3) is the double cover of SO(3).

The preimage of [3,3,5]+ in Spin(4) (a four-dimensional analogue of 2I) is precisely the product group 2I × 2I of order 14400.

The coset space Spin(3) / 2I = S3 / 2I is a spherical 3-manifold called the Poincaré homology sphere.

The fundamental group of the Poincaré sphere is isomorphic to the binary icosahedral group, as the Poincaré sphere is the quotient of a 3-sphere by the binary icosahedral group.