Binary octahedral group

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 and all possible sign combinations. All 48 elements have absolute value 1 and therefore lie in the unit quaternion group Sp(1). ==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. 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. Quaternion generators with these relations are given by :r = \tfrac{1}{\sqrt 2}(i+j) \qquad s = \tfrac{1}{2}(1+i+j+k) \qquad t = \tfrac{1}{\sqrt 2}(1+i), with r^2 = s^3 = t^4 = rst = -1. Subgroups • ⟨p⟩≃Z2p, (p)≃Zp (cyclic groups) 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. There are also isomorphic copies of the binary dihedral groups Q8 and Q12 in 2O. All other subgroups are cyclic groups generated by the various elements (with orders 3, 4, 6, and 8). ==Higher dimensions==

The binary octahedral group generalizes to higher dimensions: just as the octahedron generalizes to the orthoplex, the octahedral group in SO(3) generalizes to the hyperoctahedral group in SO(n), which has a binary cover under the map \operatorname{Spin}(n) \to SO(n). ==See also==