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In mathematics, the binary octahedral group, name as 2O or ⟨2,3,4⟩ is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group O or of order 24 by a cyclic group of order 2, and is the preimage of the octahedral group under the 2:1 covering homomorphism Spin → SO {\displaystyle \operatorname {Spin} \to \operatorname {SO} } of the special orthogonal group by the spin group. It follows that the binary octahedral group is a discrete subgroup of Spin of order 48.
The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin ≅ Sp {\displaystyle \operatorname {Spin} \cong \operatorname {Sp} } where Sp is the multiplicative group of unit quaternions.