Binary octahedral group

In mathematics, the binary octahedral group, name as 2O or is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group O or (2,3,4) of order 24 by a cyclic group of order 2, and is the 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.) 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 and all possible sign combinations. All 48 elements have absolute value 1 and therefore lie in the unit quaternion group Sp(1). The binary octahedral group, denoted by 2O, fits into the short exact sequence 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. The group 2O has a presentation given by or equivalently, Quaternion generators with these relations are given by with The binary tetrahedral group, 2T, consisting of the 24 Hurwitz units, forms a normal subgroup of index 2. The quaternion group, Q8, 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). These two groups, together with the center {±1}, are the only nontrivial normal subgroups of 2O. The generalized quaternion group, Q16, also forms a subgroup of 2O, index 3. This subgroup is self-normalizing so its conjugacy class has 3 members.

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Related concepts (8)

Binary cyclic group

In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, , thought of as an extension of the cyclic group by a cyclic group of order 2. Coxeter writes the binary cyclic group with angle-brackets, ⟨n⟩, and the index 2 subgroup as (n) or [n]+. It is the binary polyhedral group corresponding to the cyclic group. In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations () under the 2:1 covering homomorphism of the special orthogonal group by the spin group.

Binary tetrahedral group

In mathematics, the binary tetrahedral group, denoted 2T or , is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order 2, and is the of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the special orthogonal group by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3 by G.C.