In mathematics, the binary octahedral group, name as 2O or ⟨2,3,4⟩[1] 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 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.)
^ Coxeter&Moser: Generators and Relations for discrete groups: <l,m,n>: Rl = Sm = Tn = RST
and 20 Related for: Binary octahedral group information
symmetry Icosahedral symmetry Binaryoctahedralgroup Hyperoctahedral group Learning materials related to Full octahedralgroup at Wikiversity John Conway...
mathematics, the binary icosahedral group 2I or ⟨2,3,5⟩ is a certain nonabelian group of order 120. It is an extension of the icosahedral group I or (2,3,5)...
of the small groups that do not have a normal p-complement include: Order 24: The symmetric group S4 Order 48: The binaryoctahedralgroup and the product...
multiplication as the group operation. binary dihedral group, ⟨2,2,n⟩, order 4n binary tetrahedral group, ⟨2,3,3⟩, order 24 binaryoctahedralgroup, ⟨2,3,4⟩, order...
multiply as the binaryoctahedralgroup. Another subgroup is formed by 120 icosians which multiply in the manner of the binary icosahedral group. A hyperbolic...
symmetric group, 2⋅An → An, with the alternating group being the (rotational) symmetry group of the n-simplex. higher binaryoctahedralgroups, corresponding...
(2.95 Å) Sn-P bonds. The result is that Sn(II) adopts highly distorted octahedral geometry. The structure of tin triphosphide resembles that of gray arsenic...
as a snub tetrahedron has chiral symmetry. Octahedral symmetry Icosahedral symmetry Binary tetrahedral group Koca et al. 2016. Peter R. Cromwell, Polyhedra...
hydrogen atoms randomly occupy the octahedral interstices in the metal lattice (in an fcc lattice there is one octahedral hole per metal atom). The limit...
explains the name dihedral group (in analogy to tetrahedral, octahedral and icosahedral group, referring to the proper symmetry groups of a regular tetrahedron...
not an anion. These hydrogen compounds can be grouped into several types. Binary hydrogen compounds in group 1 are the ionic hydrides (also called saline...
molecular geometry of binary hexafluorides is generally octahedral, although some derivatives are distorted from Oh symmetry. For the main group hexafluorides...
a group. In mathematics, a group is a set with a binary operation that satisfies the four properties listed below. In a symmetry group, the group elements...
halides are known. The binary tetrahalides can be converted to the hexahalides [TcX6]2− (X = F, Cl, Br, I), which adopt octahedral molecular geometry. More...
subgroups of S U ( 2 ) {\displaystyle SU(2)} , the binary polyhedral groups; properly, binary polyhedral groups correspond to the simply laced affine Dynkin...
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