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| In [[mathematics]], the '''binary octahedral group''', name as 2O or <2,3,4> is a certain [[nonabelian group]] of [[order (group theory)|order]] 48. It is an [[group extension|extension]] of the [[octahedral group]] ''O'' or (2,3,4) of order 24 by a [[cyclic group]] of order 2, and is the [[preimage]] of the octahedral group under the 2:1 [[covering homomorphism]] <math>\operatorname{Spin}(3) \to \operatorname{SO}(3)</math> of the [[special orthogonal group]] by the [[spin group]]. It follows that the binary octahedral group is a [[discrete subgroup]] of Spin(3) of order 48.
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| The binary octahedral group is most easily described concretely as a discrete subgroup of the unit [[quaternion]]s, under the isomorphism <math>\operatorname{Spin}(3) \cong \operatorname{Sp}(1)</math> where [[Sp(1)]] is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on [[quaternions and spatial rotation]]s.)
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| ==Elements==
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| Explicitly, the binary octahedral group is given as the union of the 24 [[Hurwitz unit]]s
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| :<math>\{\pm 1,\pm i,\pm j,\pm k,\tfrac{1}{2}(\pm 1 \pm i \pm j \pm k)\}</math>
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| with all 24 quaternions obtained from
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| :<math>\tfrac{1}{\sqrt 2}(\pm 1 \pm 1i + 0j + 0k)</math>
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| 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).
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| ==Properties==
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| The binary octahedral group, denoted by 2''O'', fits into the [[short exact sequence]]
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| :<math>1\to\{\pm 1\}\to 2O\to O \to 1.\,</math> | |
| This sequence does not [[split exact sequence|split]], meaning that 2''O'' is ''not'' a [[semidirect product]] of {±1} by ''O''. In fact, there is no subgroup of 2''O'' isomorphic to ''O''.
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| The [[center of a group|center]] of 2''O'' is the subgroup {±1}, so that the [[inner automorphism group]] is isomorphic to ''O''. The full [[automorphism group]] is isomorphic to ''O'' × '''Z'''<sub>2</sub>.
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| ===Presentation===
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| The group 2''O'' has a [[group presentation|presentation]] given by
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| :<math>\langle r,s,t \mid r^2 = s^3 = t^4 = rst \rangle</math>
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| or equivalently,
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| :<math>\langle s,t \mid (st)^2 = s^3 = t^4 \rangle.</math>
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| Generators with these relations are given by
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| :<math>s = -\tfrac{1}{2}(1+i+j+k) \qquad t = \tfrac{1}{\sqrt 2}(1+i).</math>
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| ===Subgroups===
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| The [[quaternion group]] consisting of the 8 [[Lipschitz unit]]s forms a [[normal subgroup]] of 2''O'' of [[index (group theory)|index]] 6. The [[quotient group]] is isomorphic to ''S''<sub>3</sub> (the [[symmetric group]] on 3 letters). The [[binary tetrahedral group]], consisting of the 24 [[Hurwitz unit]]s, forms a normal subgroup of index 2. These two groups, together with the center {±1}, are the only nontrivial normal subgroups of 2''O''.
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| The [[generalized quaternion group]] of order 16 also forms a subgroup of 2''O''. This subgroup is [[self-normalizing]] so its [[conjugacy class]] has 3 members. There are also isomorphic copies of the [[binary dihedral group]]s of orders 8 and 12 in 2''O''. All other subgroups are [[cyclic group]]s generated by the various elements (with orders 3, 4, 6, and 8).
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| ==Higher dimensions==
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| The binary octahedral group generalizes to higher dimensions: just as the octahedron generalizes to the [[hyperoctahedron]], the octahedral group in SO(3) generalizes to the [[hyperoctahedral group]] in SO(''n''), which has a binary cover under the map <math>\operatorname{Spin}(n) \to SO(n).</math>
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| ==See also==
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| *[[binary polyhedral group]]
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| *[[binary cyclic group]]
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| *[[binary dihedral group]]
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| *[[binary tetrahedral group]]
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| *[[binary icosahedral group]]
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| *[[hyperoctahedral group]]
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| ==References==
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| *{{cite book | author=Coxeter, H. S. M. and Moser, W. O. J. | title=Generators and Relations for Discrete Groups, 4th edition | location=New York | publisher=Springer-Verlag | year=1980 | isbn=0-387-09212-9}}
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| *{{cite book | first = John H. | last = Conway | coauthors = Smith, Derek A. | authorlink = John Horton Conway | title = On Quaternions and Octonions | publisher = AK Peters, Ltd | location = Natick, Massachusetts | year = 2003 | isbn = 1-56881-134-9}}
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| [[Category:Binary polyhedral groups|Octahedral]]
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