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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 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 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. The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.) ==Elements== Explicitly, the binary octahedral group is given as the union of the 24 Hurwitz units : with all 24 quaternions obtained from : 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「binary octahedral group」の詳細全文を読む スポンサード リンク
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