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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
Binary_octahedral_group
3D symmetry group
symmetry Binary octahedral group Hyperoctahedral group 25 great circles of the spherical octahedron Learning materials related to Full octahedral group at Wikiversity
Octahedral_symmetry
Nonabelian group in algebraic group theory
Binary dihedral group, ⟨2,2,n⟩, order 4n Binary octahedral group, 2O = ⟨2,3,4⟩, order 48 Binary icosahedral group, 2I = ⟨2,3,5⟩, order 120 Coxeter, Complex
Binary_tetrahedral_group
Nonabelian group of order 120
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)
Binary_icosahedral_group
Groups of point isometries in 3 dimensions
binary tetrahedral group, ⟨2,3,3⟩, order 24 E 7 {\displaystyle E_{7}} : binary octahedral group, ⟨2,3,4⟩, order 48 E 8 {\displaystyle E_{8}} : binary
Point groups in three dimensions
Point_groups_in_three_dimensions
Type of cyclic group in group theory
polyhedral group binary cyclic group, ⟨n⟩, order 2n binary tetrahedral group, 2T = ⟨2,3,3⟩, order 24 binary octahedral group, 2O = ⟨2,3,4⟩, order 48 binary icosahedral
Dicyclic_group
Algebraic structure
multiplication as the group operation. binary dihedral group, ⟨2,2,n⟩, order 4n binary tetrahedral group, ⟨2,3,3⟩, order 24 binary octahedral group, ⟨2,3,4⟩, order
Binary_cyclic_group
of the small groups that do not have a normal p-complement include: Order 24: The symmetric group S4 Order 48: The binary octahedral group and the product
List_of_small_groups
In group theory, equivalence class under the relation of conjugation
"Inner-automorphism between the octahedral group and the binary octahedral group". Conrad, Keith. "Dihedral groups" (PDF). Retrieved 3 September 2025
Conjugacy_class
Concept in geometry
U1(H): the binary cyclic groups, binary dihedral groups, binary tetrahedral group, binary octahedral group, and binary icosahedral group. Hoggar, S.G
Quaternionic_polytope
Subclass of manifold
3, and 3. The fundamental group is a product of a cyclic group of order m coprime to 6 with the binary octahedral group (of order 48) which has the
Spherical_3-manifold
Regular object in four dimensional geometry
2020, p. 78, §4.2.5. Egan 2021; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations. Koca, Al-Ajmi & Koc
24-cell
Quaternion of norm 1 (unit quaternion)
multiply as the binary octahedral group. Another subgroup is formed by 120 icosians which multiply in the manner of the binary icosahedral group. A hyperbolic
Versor
Use of mathematical groups in magnetochemistry
the cyclic groups, the binary dihedral groups, the binary tetrahedral group, the binary octahedral group, and the binary icosahedral group; and for SO(3)
Finite_subgroups_of_SU(2)
Algebraic structure
Zassenhaus's paper. The binary tetrahedral, octahedral and icosahedral groups are central extensions of the rotational symmetry groups of the platonic solids;
Near-field_(mathematics)
Double cover Lie group of the special orthogonal group
symmetric group, 2⋅An → An, with the alternating group being the (rotational) symmetry group of the n-simplex. higher binary octahedral groups, corresponding
Spin_group
Chemical compound
of the strongest oxidants, capable of oxidising xenon and O2. PtF6 is octahedral in both the solid state and in the gaseous state. The Pt-F bond lengths
Platinum_hexafluoride
Symmetry of molecules of chemical compounds
equally applicable to other octahedral groups because O and Td are distinct examples of the same abstract group. W(CO)6 has octahedral geometry. The irreducible
Molecular_symmetry
Mathematical classification
subgroups of S U ( 2 ) {\displaystyle SU(2)} , the binary polyhedral groups; properly, binary polyhedral groups correspond to the simply laced affine Dynkin
ADE_classification
Chemical compound
close packing. In liquid and gas phase, MoF6 adopt octahedral molecular geometry with point group Oh. The Mo–F bond length is 1.817 Å. Molybdenum hexafluoride
Molybdenum_hexafluoride
Group of symmetries of a regular polygon
explains the name dihedral group (in analogy to tetrahedral, octahedral and icosahedral group, referring to the proper symmetry groups of a regular tetrahedron
Dihedral_group
Chemical compound
(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
Tin_triphosphide
Group of chemical 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
Group_7_element
Recursive integer sequence
binary operator can be represented in terms of a full binary tree, by labeling each leaf a, b, c, d. It follows that Cn is the number of full binary trees
Catalan_number
Natural number
26015 = number of partitions of 38 26214 = octahedral number 26227 = cuban prime 26272 = number of 20-bead binary necklaces with beads of 2 colors where the
20,000
projected to the unit 3-sphere. These 48 vertices correspond to the binary octahedral group 2O or <2,3,4>, order 48. Thus, the 288-cell is the only non-regular
Truncated_24-cells
Chemical compound
metal. In this compound, the hafnium centers adopt octahedral coordination geometry. Like most binary metal halides, the compound is a polymeric. It is
Hafnium(IV)_iodide
Natural number
2047 – super-Poulet number, Woodall number, decagonal number, a centered octahedral number, 2047 = 211 – 1 = 23 × 89 and is the first Mersenne number that
2000_(number)
Set with associative invertible operation
the three axioms. Formally, a group is an ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the
Group_(mathematics)
Chemical compounds containing only hydrogen and one other chemical element
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
Binary_compounds_of_hydrogen
Chemical compound
that of ruthenium pentafluoride. Within the tetramers, each Pt adopts octahedral molecular geometry, with two bridging fluoride ligands. Bartlett, N.;
Platinum_pentafluoride
Molecule with a hydrogen bound to a more electropositive element or group
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
Hydride
Chemical compound
form important for the liquid or gas phase) has octahedral molecular geometry, which has point group (Oh). The Ru–F bond length is 1.818 Å. CRC Handbook
Ruthenium_hexafluoride
Chemical compound
hygroscopic crystalline solid and is one of two fluorides of tellurium. The other binary fluoride is tellurium hexafluoride. The widely reported Te2F10 has been
Tellurium_tetrafluoride
3D symmetry group
as a snub tetrahedron has chiral symmetry. Octahedral symmetry Icosahedral symmetry Binary tetrahedral group Koca et al. 2016. Peter R. Cromwell, Polyhedra
Tetrahedral_symmetry
Chemical compound
formula PtF 4. In the solid state, the compound features platinum(IV) in octahedral coordination geometry. Platinum tetrafluoride vapour at 298.15 K consists
Platinum_tetrafluoride
Natural number
10395 = double factorial of 11 10416 = square pyramidal number 10425 = octahedral number 10430 = weird number 10433 = palindromic prime in base 44 (5H544)
10,000
Index of chemical compounds with the same name
molecular geometry of binary hexafluorides is generally octahedral, although some derivatives are distorted from Oh symmetry. For the main group hexafluorides
Hexafluoride
Chemical compound
7-coordinate group of O2− atoms, the oxygen ions are in an octahedral shape around the metal atom, and there is one oxygen ion above one of the octahedral faces
Lanthanum_oxide
Chemical compound
form important for the liquid or gas phase) has octahedral molecular geometry, which has point group (Oh). The Ir–F bond length is 1.833 Å. Calculations
Iridium_hexafluoride
Numbers obtained by adding the two previous ones
different ways among the set of binary strings, or equivalently, among the subsets of a given set. The number of binary strings of length n without consecutive
Fibonacci_sequence
Figurate number
conditional branch if implemented as t = (n|1) * ((n+1)/2). If n is odd, the binary OR operation n|1 has no effect, so this is equivalent to t = n * ((n+1)/2)
Triangular_number
Chemical compound
isostructural platinum pentafluoride. Within the tetramers, each Ru adopts octahedral molecular geometry, with two bridging fluoride ligands. Ruthenium pentafluoride
Ruthenium_pentafluoride
Chemical compound
form important for the liquid or gas phase) has octahedral molecular geometry, which has point group (Oh). The Re–F bond length is 1.823 Å. Rhenium hexafluoride
Rhenium_hexafluoride
relatively stable and crystalizeses as a near perfect octahedral. In contrast to V(IV), the neutral binary V(V) could not be synthesized and attempts result
Transition metal azide complex
Transition_metal_azide_complex
Number used for counting
done even where the distinction is relevant, as with binary numerals which are often called "binary numbers". Natural numbers are used for counting and
Natural_number
Chemical compound
form important for the liquid or gas phase) has octahedral molecular geometry, which has point group (Oh). The Os–F bond length is 1.827 Å. Partial hydrolysis
Osmium_hexafluoride
Chemical compound
a lattice, and X-ray crystallography shows that its crystals form an octahedral Prussian-blue-like structure. Bergstrom, F. W. (July 1924). "The Reaction
Aluminium_cyanide
Chemical compound with formula RhF6
rhodium(VI) compound. It is one of seventeen known binary hexafluorides. The RhF6 molecule has octahedral molecular geometry. Consistent with its d3 configuration
Rhodium_hexafluoride
Group of elements in the periodic table
dienes. Like all binary halides, those of vanadium are Lewis acidic, especially those of V(IV) and V(V). Many of the halides form octahedral complexes with
Group_5_element
Iterative algorithm on numbers
their distribution. In 2024, Haruo Iwasaki of the Ranzan Mathematics Study Group (headed by Kenichi Iyanaga) showed that in order for a natural number to
Kaprekar's_routine
by the 24-element octahedral group isomorphic to the rotations of a cube. The elements of the set are permutations and the binary operator is composition
Nomos_Alpha
3D symmetry group
but which is not homotopic to it. Tetrahedral symmetry Octahedral symmetry Binary icosahedral group Icosian calculus Sir William Rowan Hamilton (1856), "Memorandum
Icosahedral_symmetry
SiH4 and HN3. The hexaazidosicalte salt [(Ph3P)2N]2[Si(N3)6] adopts an octahedral molecular geometry, a very rare case of silicon in an N6 environment.
Main_group_azido_compounds
Number that remains the same when its digits are reversed
numeral systems other than decimal. For example, the binary palindromic numbers are those with the binary representations: 0, 1, 11, 101, 111, 1001, 1111,
Palindromic_number
Ion or molecule bound to a metal atom
d-orbitals, whereas in an octahedral complex the d-orbitals are influenced by 6 ligands. When the coordination number is neither octahedral nor tetrahedral, the
Ligand
Group of highly reactive chemical elements
an octahedral hexaaqua ion. There are also probably six water molecules in the primary solvation sphere of the sodium ion, forming the octahedral [Na(H2O)6]+
Alkali_metal
Product of two prime numbers
signal aimed at a star cluster. It consisted of 1679 {\displaystyle 1679} binary digits intended to be interpreted as a 23 × 73 {\displaystyle 23\times 73}
Semiprime
Chemical compounds containing zinc
is found in metallo-enzymes such as carbonic anhydrase. Six-coordinate octahedral complexes are also common, such as the aquo complex [Zn(H2O)6]2+, which
Zinc_compounds
Natural number
safe prime 7103 – Sophie Germain prime, sexy prime with 7109 7106 – octahedral number 7109 – super-prime, sexy prime with 7103 7121 – Sophie Germain
7000_(number)
Type of chemical compound
dienes. Like all binary halides, those of vanadium are Lewis acidic, especially those of V(IV) and V(V). Many of the halides form octahedral complexes with
Vanadium_compounds
an octahedral complex chromium tris(diethyldithiocarbamate), an octahedral complex manganese tris(dimethyldithtiocarbamate), an octahedral complex
Transition metal dithiocarbamate complexes
Transition_metal_dithiocarbamate_complexes
Chemical compound
perfect octahedral symmetry, and indeed electron diffraction combined with high-level calculations indicate that the compound's point group is C3v. It
Xenon_hexafluoride
Chemical compound
manganese(II) halide at ~250 °C. Like vanadium(III) fluoride, MnF3 features octahedral metal centers with the same average M-F bond distances. In the Mn compound
Manganese(III)_fluoride
Geometrical structure
close-packed arrangement, and then arrange the small spheres within the octahedral and tetrahedral gaps. The density of this interstitial packing depends
Sphere_packing
Chemical compound
polymeric structure. The Re–Re bonding distance is 2.728 Å. Re centers are octahedral, being surrounded by six chloride ligands. Pairs of octahedra share faces
Rhenium(IV)_chloride
of order 33, product of first four odd primes (3*5*7*11) 1156 = 342, octahedral number, centered pentagonal number, centered hendecagonal number. 1157
1000_(number)
Two raised to an integer power
Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form
Power_of_two
Class of elements
The salts NaTlF4 and Na3TlF6 do not contain discrete tetrahedral and octahedral anions. The structure of NaTlF4 is the same as fluorite (CaF2) with NaI
Thallium_halides
Group of chemical elements
tetrahedral species, such as MCl2− 4. Both zinc and cadmium can also form octahedral complexes such as the aqua ions [M(H2O)6]2+ which are present in aqueous
Group_12_element
Union of crystal groups with related structures and lattices
be described as an HCP lattice of arsenic with nickel occupying each octahedral void. Compounds adopting the NiAs structure are generally the chalcogenides
Hexagonal_crystal_family
Crystallographic system where the unit cell is in the shape of a cube
so as to form a 3D checkerboard pattern. The rock-salt structure has octahedral coordination: Each atom's nearest neighbors consist of six atoms of the
Cubic_crystal_system
Oxide mineral
slightly distorted hexagonal close packing, in which two-thirds of the octahedral sites between the oxygen ions are occupied by aluminium ions. The absence
Corundum
Chemical element with atomic number 77 (Ir)
Iridium hexafluoride, IrF 6, is a volatile yellow solid, composed of octahedral molecules. It decomposes in water and is reduced to IrF 4. Iridium pentafluoride
Iridium
Number equal to the sum of its proper divisors
{\displaystyle 2^{p-1}(2^{p}-1),} every even perfect number is represented in binary form as p ones followed by p − 1 zeros; for example: 6 10 = 2 2 + 2 1 =
Perfect_number
Number that can be used to count certain kinds of binary trees
Etherington and Joseph Wedderburn that can be used to count certain kinds of binary trees. The first few numbers in the sequence are 0, 1, 1, 1, 2, 3, 6, 11
Wedderburn–Etherington_number
Centered figurate number representing an octahedron
In mathematics, a centered octahedral number or Haüy octahedral number is a figurate number that counts the points of a three-dimensional integer lattice
Centered_octahedral_number
Arithmetic operation
superscript to the right of the base as bn or in computer code as b^n. This binary operation is often read as "b to the power n"; it may also be referred to
Exponentiation
Chemical compound
"interconnect". It is one of seventeen known binary hexafluorides. The WF6 molecule is octahedral with the symmetry point group of Oh. The W–F bond distances are
Tungsten_hexafluoride
Chemical compounds containing technetium
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
Technetium_compounds
Chemical compound and greenhouse gas
a colorless, odorless, non-flammable, and non-toxic gas. SF 6 has an octahedral geometry, consisting of six fluorine atoms attached to a central sulfur
Sulfur_hexafluoride
Chemical compound involving ionic bonding
close-packed arrangements of spheres, with the cations occupying tetrahedral or octahedral interstices. Depending on the stoichiometry of the salt, and the coordination
Salt_(chemistry)
Number divisible only by 1 and itself
p^{n}} . By Lagrange's theorem, any group of prime order is a cyclic group, and by Burnside's theorem any group whose order is divisible by only two
Prime_number
Solid with six equal square faces
symmetry point group. In this case, the dual polyhedron of a cube is the regular octahedron, and both of these polyhedra have the same octahedral symmetry.
Cube
occupying 3/2 of the 4 octahedral sites. The second type of sub unit cell has only 3+ cations occupying 5/2 of the 4 octahedral sites. These two types
Siegenite
Iridium hexafluoride, IrF 6, is a volatile yellow solid, composed of octahedral molecules. It decomposes in water and is reduced to IrF 4,. Iridium pentafluoride
Iridium_compounds
Count of permutations by cycles
4 2 ] = 11 {\displaystyle \left[{4 \atop 2}\right]=11} : the symmetric group on 4 objects has 3 permutations of the form ( ∙ ∙ ) ( ∙ ∙ ) {\displaystyle
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Generalized sphere of dimension n (mathematics)
shape of a cross-polytope. The octahedral 1 {\displaystyle 1} -sphere is a square (without its interior). The octahedral 2 {\displaystyle 2} -sphere
N-sphere
Field of chemistry
titanium (e.g., TiCl4) to square planar for some nickel complexes to octahedral for coordination complexes of cobalt. A range of transition metals can
Inorganic_chemistry
Chemical compound
size between potassium (144 pm) and rubidium ions (152 pm) when in an octahedral environment. The vibrational spectrum consists of two main sets of absorptions
Ammonium
Number n where n and totient(n) are coprime
An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic. Any prime number is clearly cyclic. All cyclic numbers
Cyclic_number_(group_theory)
Chemical compound
platinum tetrabromide and technetium tetrachloride. As such, osmium is in octahedral coordination. Each osmium center bonds to four doubly bridging bromide
Osmium_tetrabromide
Natural number
stella octangula, or eight-pointed star, is the only stellation with octahedral symmetry. It has eight triangular faces alongside eight vertices that
8
Pair of integers related by their divisors
divided into two groups. The first three lemmas deal with the determination of the aliquot parts of a natural integer. The second group of lemmas deals
Amicable_numbers
Chemical compounds containing iodine
only the expected tetrahedral IO− 4, but also square-pyramidal IO3− 5, octahedral orthoperiodate IO5− 6, [IO3(OH)3]2−, [I2O8(OH2)]4−, and I 2O4− 9. They
Iodine_compounds
Integer having only small prime factors
algorithm for computing discrete logarithms has a running time of O(n1/2)—for groups of n-smooth order. Moreover, m is said to be smooth over a set A if there
Smooth_number
Chemical element with atomic number 43 (Tc)
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
Technetium
Chemical compound
is the highest fluoride of neptunium, and is also one of seventeen known binary hexafluorides. It is a volatile orange crystalline solid. It is relatively
Neptunium(VI)_fluoride
Chemical compound
and Te.[citation needed] Tellurium hexafluoride is a highly symmetric octahedral molecule. Its physical properties resemble those of the hexafluorides
Tellurium_hexafluoride
Chemical compound
parameters". Non-Tetrahedrally Bonded Elements and Binary Compounds I. Landolt-Börnstein - Group III Condensed Matter. Vol. 41C. 1998. pp. 1–8. doi:10
Tin_telluride
BINARY OCTAHEDRAL-GROUP
BINARY OCTAHEDRAL-GROUP
Boy/Male
Irish
An ancient Irish name whos meaning is lost in antiquety.
Girl/Female
English
Originally a diminutive used for names ending in -bina, like Albina, Columbina, and Robina, now...
Girl/Female
Indian
Modesty
Female
Hebrew
(×‘Ö¼Ö´×™× Ö¸×”) Hebrew name BINA means "intelligence, wisdom."Â
Surname or Lastname
English
English : variant spelling of Vickery.
Girl/Female
Indian
(the wife of Sage Kashyap)
Female
English
English pet form of German Belinda, possibly BINDY means "bright serpent" or "bright linden tree."
Boy/Male
Latin
Happy; Cheerful.
Girl/Female
Hindu
Shore, Musical instrument, Goddess of wealth
Male
Hindi/Indian
(विनय) Hindi name VINAY means "leading asunder."
Boy/Male
Indian
An intimate particle of the God of heaven
Male
English
English unisex form of Latin Hilarius and Hilaria, HILARY means "joyful; happy."Â Originally, this was strictly a masculine name.
Female
Turkish
Turkish name PINAR means "spring."
Girl/Female
Hindu
Shore, Musical instrument, Goddess of wealth
Boy/Male
American, Australian, French, German, Greek, Latin, Polish, Swedish
Cheerful; Happy; Joyful; Similar to Hilary
Male
Hindi/Indian
Variant spelling of Hindi Vijay, BIJAY means "victory."
Female
Hebrew
Variant spelling of Hebrew Bina, BINAH means "intelligence, wisdom."Â
Male
Scandinavian
Scandinavian form of Old Norse Einarr, EINAR means "lone warrior."
Boy/Male
Indian, Punjabi, Sikh
Blessing
Surname or Lastname
English (chiefly South Yorkshire)
English (chiefly South Yorkshire) : topographic name for someone who lived on land enclosed by a bend in a river, from Old English binnan ēa ‘within the river’, or a habitational name from places in Kent called Binney and Binny, which have this origin.Scottish : habitational name from Binney or Binniehill near Falkirk, named in Gaelic as Beinnach, from beinn ‘hill’ + the locative suffix -ach.
BINARY OCTAHEDRAL-GROUP
BINARY OCTAHEDRAL-GROUP
Boy/Male
Tamil
Bull
Girl/Female
Tamil
Thirst
Boy/Male
American, British, English
Divine Warrior
Boy/Male
Hindu, Indian
Rice Put During Pooja in Hindu Rituals
Girl/Female
Latin American Spanish
Dazzling white. Bright, glowing white. Derived from the feminine of the Latin adjective meaning...
Boy/Male
Hindu
Lord Shiva
Boy/Male
Tamil
Lord of law or one who is well versed in law, Name of Lord Shiva
Boy/Male
Bengali, Hindu, Indian, Kannada, Telugu, Traditional
Moon-crested Lord
Girl/Female
Hindu
Destroyer of enemies, Star
Girl/Female
Tamil
BINARY OCTAHEDRAL-GROUP
BINARY OCTAHEDRAL-GROUP
BINARY OCTAHEDRAL-GROUP
BINARY OCTAHEDRAL-GROUP
BINARY OCTAHEDRAL-GROUP
a.
Presenting a combination of a cube and an octahedron.
a.
Of a pale yellowish color; as, Canary stone.
a.
See Octahedral.
a.
Containing ten; tenfold; proceeding by tens; as, the denary, or decimal, scale.
a.
See Octahedral.
n.
A combination of a cube and octahedron, esp. one in which the octahedral faces meet at the middle of the cubic edges.
a.
Relating or belonging to bile; conveying bile; as, biliary acids; biliary ducts.
v. i.
To perform the canary dance; to move nimbly; to caper.
a.
Having eight faces or sides; of, pertaining to, or formed in, octahedrons; as, octahedral cleavage.
a.
Resembling the aisles of a cathedral; as, cathedral walks.
n.
A binary compound of silicon, or one regarded as binary.
n.
A pale yellow color, like that of a canary bird.
n.
Wine made in the Canary Islands; sack.
a.
Of or pertaining to the Canary Islands; as, canary wine; canary birds.
n.
A solid bounded by eight faces. The regular octahedron is contained by eight equal equilateral triangles.
a.
Of or pertaining to the urine; as, the urinary bladder; urinary excretions.
a.
lasting for one day; as, a diary fever.
n.
See Finery.
a.
Pertaining to the head church of a diocese; as, a cathedral church; cathedral service.
n.
A canary bird.