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 perfect central extension of a simple group.

Explicitly, it fits into the short exact sequence

${\displaystyle 1\to \{\pm 1\}\to 2I\to I\to 1.\,}$

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 ${\displaystyle I\cong A_{5}}$ - any automorphism of 2I fixes the non-trivial element of the center (${\displaystyle -1}$), 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).

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: ${\displaystyle H_{1}(2I;\mathbf {Z} )\cong H_{2}(2I;\mathbf {Z} )\cong 0.}$ 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). In fact, the binary icosahedral group is the smallest (non-trivial) superperfect group.

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

Isomorphisms

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). Note that the symmetric group ${\displaystyle S_{5}}$ does have a 4-dimensional representation (its usual lowest-dimensional irreducible representation as the full symmetries of the ${\displaystyle (n-1)}$-simplex), and that the full symmetries of the 4-simplex are thus ${\displaystyle S_{5},}$ not the full icosahedral group (these are two different groups of order 120).

The binary icosahedral group can be considered as the double cover of the alternating group ${\displaystyle A_{5},}$ denoted ${\displaystyle 2\cdot A_{5}\cong 2I;}$ this isomorphism covers the isomorphism of the icosahedral group with the alternating group ${\displaystyle A_{5}\cong I,}$. Just as ${\displaystyle I}$ is a discrete subgroup of ${\displaystyle \mathrm {SO} (3)}$, ${\displaystyle 2I}$ is a discrete subgroup of the double over of ${\displaystyle \mathrm {SO} (3)}$, namely ${\displaystyle \mathrm {Spin} (3)\cong \mathrm {SU} (2)}$. The 2-1 homomorphism from ${\displaystyle \mathrm {Spin} (3)}$ to ${\displaystyle \mathrm {SO} (3)}$ then restricts to the 2-1 homomorphism from ${\displaystyle 2I}$ to ${\displaystyle I}$.

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 ${\displaystyle I\cong A_{5}}$ with the projective special linear group PSL(2,5).

Note also the exceptional isomorphism ${\displaystyle PGL(2,5)\cong S_{5},}$ which is a different group of order 120, with the commutative square of SL, GL, PSL, PGL being isomorphic to a commutative square of ${\displaystyle 2\cdot A_{5},2\cdot S_{5},A_{5},S_{5},}$ which are isomorphic to subgroups of the commutative square of Spin(4), Pin(4), SO(4), O(4).

Presentation

The group 2I has a presentation given by

${\displaystyle \langle r,s,t\mid r^{2}=s^{3}=t^{5}=rst\rangle }$

or equivalently,

${\displaystyle \langle s,t\mid (st)^{2}=s^{3}=t^{5}\rangle .}$

Generators with these relations are given by

${\displaystyle s={\tfrac {1}{2}}(1+i+j+k)\qquad t={\tfrac {1}{2}}(\varphi +\varphi ^{-1}i+j).}$

Subgroups

Subgroups:
• binary tetrahedral group: 2T=⟨2,3,3⟩
• 3 binary dihedral groups: Q₂₀=⟨2,2,5⟩, Q₁₂=⟨2,2,3⟩, Q₈=⟨2,2,2⟩
• 3 binary cyclic groups: Z₁₀=⟨5⟩, Z₆=⟨3⟩, Z₄=⟨2⟩
• 3 cyclic groups: Z₅=(5), Z₃=(3), Z₂=(2)
• 1 trivial group: ( )

The only proper normal subgroup of 2I is the center { ±1 }.

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 { ±1 }.

${\displaystyle -1}$ 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 (which can have odd order), the only other subgroups of 2I (up to conjugation) are:^[2]

• 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 24 Hurwitz units form an index 5 subgroup called the binary tetrahedral group; this covers a chiral tetrahedral group. This group is self-normalizing so its conjugacy class has 5 members (this gives a map ${\displaystyle 2I\to S_{5}}$ whose image is ${\displaystyle A_{5}}$).