Binary octahedral group

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In mathematics, the binary octahedral group is an extension of the octahedral group O of order 24 by a cyclic group of order 2. It can be defined as the preimage of the octahedral group under the 2:1 covering homomorphism

\mathrm{Sp}(1) \to \mathrm{SO}(3).\,

where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.) It follows that the binary octahedral group is discrete subgroup of Sp(1) of order 48.

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[edit] Elements

Explicitly, the binary octahedral group is given as the union of the 24 Hurwitz units

\{\pm 1,\pm i,\pm j,\pm k,\tfrac{1}{2}(\pm 1 \pm i \pm j \pm k)\}

with all 24 quaternions obtained from

\tfrac{1}{\sqrt 2}(\pm 1 \pm 1i + 0j + 0k)

by a permutation of coordinates (all possible sign combinations). All 48 elements have absolute value 1 and therefore lie in the unit quaternion group Sp(1).

[edit] Properties

The binary octahedral group, denoted by 2O, fits into the short exact sequence

1\to\{\pm 1\}\to 2O\to O \to 1.\,

This sequence does not split, meaning that 2O is not a semidirect product of {±1} by O. In fact, there is no subgroup of 2O isomorphic to O.

The center of 2O is the subgroup {±1}, so that the inner automorphism group is isomorphic to O. The full automorphism group is isomorphic to O × Z2.

[edit] Presentation

The group 2O has a presentation given by

\langle r,s,t \mid r^2 = s^3 = t^4 = rst \rangle

or equivalently,

\langle s,t \mid (st)^2 = s^3 = t^4 \rangle.

Generators with these relations are given by

s = \tfrac{1}{2}(1+i+j+k) \qquad t = \tfrac{1}{\sqrt 2}(1+i).

[edit] Subgroups

The quaternion group consisting of the 8 Lipschitz units forms a normal subgroup of 2O of index 6. The quotient group is isomorphic to S3 (the symmetric group on 3 letters). The binary tetrahedral group, consisting of the 24 Hurwitz units, forms a normal subgroup of index 2. These two groups, together with the center {±1}, are the only nontrivial normal subgroups of 2O.

The generalized quaternion group of order 16 also forms a subgroup of 2O. This subgroup is self-normalizing so its conjugacy class has 3 members. There are also isomorphic copies of the binary dihedral groups of orders 8 and 12 in 2O. All other subgroups are cyclic groups generated by the various elements (with orders 3, 4, 6, and 8).

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