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Nonabelian group in algebraic group theory
mathematics, the binary tetrahedral group, denoted 2T or ⟨2,3,3⟩, is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2
Binary_tetrahedral_group
Nonabelian group of order 120
the binary icosahedral group. Binary polyhedral group Binary cyclic group, ⟨n⟩, order 2n Binary dihedral group, ⟨2,2,n⟩, order 4n Binary tetrahedral group
Binary_icosahedral_group
3D symmetry group
has chiral symmetry. Octahedral symmetry Icosahedral symmetry Binary tetrahedral group Koca et al. 2016. Peter R. Cromwell (1997), Polyhedra, p. 295.
Tetrahedral_symmetry
polyhedral group Binary cyclic group, ⟨n⟩, order 2n Binary dihedral group, ⟨2,2,n⟩, order 4n Binary tetrahedral group, 2T=⟨2,3,3⟩, order 24 Binary icosahedral
Binary_octahedral_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
the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group
Covering groups of the alternating and symmetric groups
Covering_groups_of_the_alternating_and_symmetric_groups
Generalization of Gaussian integers to quaternions
H is a nonabelian group of order 24 known as the binary tetrahedral group. The elements of this group include the 8 elements of Q along with the 16 quaternions
Hurwitz_quaternion
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
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
Regular object in four dimensional geometry
the group of units (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the binary tetrahedral group). The
24-cell
Natural number
identified with the 24 unit Hurwitz quaternions, which form the binary tetrahedral group. The optimal sphere packing problem has been solved in dimension
24_(number)
with its symmetry, Coxeter called as 3[3]3, isomorphic to the binary tetrahedral group, order 24. The 8 vertex coordinates of this polygon can be given
Möbius–Kantor_polygon
Subclass of manifold
the center is the tetrahedral group, equivalently, the alternating group A4. When m = 1 this group is the binary tetrahedral group. These manifolds are
Spherical_3-manifold
Mathematical concept
automorphism group has order 24, contains a normal subgroup of order 8 isomorphic to the quaternion group, and is the binary tetrahedral group. If K {\displaystyle
Supersingular_elliptic_curve
Non-abelian group of order eight
of the quaternion group) Binary tetrahedral group Clifford algebra Dicyclic group Hurwitz integral quaternion List of small groups Dean, Richard (1981)
Quaternion_group
Double cover Lie group of the special orthogonal group
are two series: higher binary tetrahedral groups, corresponding to the 2-fold cover of symmetries of the n-simplex; this group can also be considered
Spin_group
Quaternion of norm 1 (unit quaternion)
Q8, these unit Hurwitz quaternions form a group G2 of order 24 called the binary tetrahedral group. The group elements, taken as points on S3, form a 24-cell
Versor
English physicist (born 1943)
generations, and predicts bileptons; and his proposal, in 1995, of the binary tetrahedral group as a flavor symmetry. All three serve as targets of opportunity
Paul_Frampton
Construction in graph theory
the McKay graphs of these binary polyhedral groups and the extended Dynkin diagrams. For example, the binary tetrahedral group T ¯ {\displaystyle {\overline
McKay_graph
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)
Chemical compound
Deokar, Piyush; Christe, Karl O.; Haiges, Ralf (4 April 2019). "The Binary Group 4 Azide Adducts [(bpy)Ti(N3)4], [(phen)Ti(N3)4], [(bpy)2Zr(N3)4]2·bpy
Titanium_tetraazide
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)
Symmetry of molecules of chemical compounds
the tetrahedral rotational group as a subgroup. Irreps of tetrahedral and octahedral groups are also related similarly to cyclic and dihedral groups and
Molecular_symmetry
Castable objects with two distinct faces for generating random outcomes
millennium BCE, include roughly regular tetrahedral (fourfaced) dice with two vertices marked, and two vertices unmarked. When a binary lot is cast singly (as is typical
Binary_lot
Set of polygons to define the surface of a 3D model
separate file (.skeleton). .veg Vega FEM tetrahedral mesh Jernej Barbič Vega FEM Open Source. Stores a tetrahedral mesh and its material properties for FEM
Polygon_mesh
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)
3D symmetry group
all genus 2 hyperbolic surfaces. Tetrahedral symmetry Icosahedral symmetry Binary octahedral group Hyperoctahedral group 25 great circles of the spherical
Octahedral_symmetry
Four-dimensional analog of the icosahedron
from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells. The 600-cell's boundary is composed of 600 tetrahedral cells with
600-cell
Figurate number
pentagonal numbers, as well as higher-dimensional polyhedra, such as the tetrahedral numbers. Taking T 0 = 0 {\displaystyle T_{0}=0} (see empty sum), the
Triangular_number
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 compounds containing zinc
is bound tetrahedrally bound to four ligands (oxide and sulfide, respectively). Many complexes, such as ZnCl42−, are tetrahedral. Tetrahedrally coordinated
Zinc_compounds
Class of chemical compounds
and related derivatives where organic groups are attached to one or more O or S. Thiophosphates feature tetrahedral phosphorus(V) centers. Organic thiophosphates
Thiophosphate
Natural number
Sophie Germain prime 7140 – triangular number, also a pronic number, tetrahedral number 7141 – sum of the first 58 primes, star number 7151 – Sophie Germain
7000_(number)
Any inorganic compound consisting of copper and sulfur atoms only
with both trigonal as well as distorted tetrahedral coordination and are rather mobile. Therefore, this group of copper sulfides shows ionic conductivity
Copper_sulfide
Class of chemical compounds
that includes compounds with Si−H and other Si−X bonds. All contain tetrahedral silicon and terminal hydrides. They only have Si−H and Si−Si single bonds
Binary silicon-hydrogen compounds
Binary_silicon-hydrogen_compounds
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
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
all of which feature tetrahedral zinc centers and bridging azide ligands. α-Zn(N3)2 crystallizes in the monoclinic space group and is stable, while the
Zinc_azide
Group of chemical elements
molecular binary metal oxide. Other examples are ruthenium(VIII) oxide and osmium(VIII) oxide. It adopts a centrosymmetric corner-shared bi-tetrahedral structure
Group_7_element
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
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
Any chemical compound having at least one boron atom
charge of the tetrahedral borate center is balanced by metal cations in the minerals, such as the sodium (Na+) in borax. The tourmaline group of borate-silicates
Boron_compounds
Group of highly reactive chemical elements
number for Li+ has been experimentally determined to be 4, forming the tetrahedral [Li(H2O)4]+: while solvation numbers of 3 to 6 have been found for lithium
Alkali_metal
(February 2019). "Coordination chemistry of organometallic or inorganic binary group 14/16 units towards d-block and f-block metal atoms". Coordination Chemistry
Chalcogenidotetrelate
Group of chemical elements
mercury many of its distinctive properties. All three metal ions form many tetrahedral species, such as MCl2− 4. Both zinc and cadmium can also form octahedral
Group_12_element
oxygen atoms and have a tetrahedral coordination. As such, Goldschmidt concluded from this that only cations with tetrahedral coordination would form
Glass_formation
Any binary chemical compound containing just silicon and another chemical element
containing silicon bonded to a more electropositive element. Binary silicon compounds can be grouped into several classes. Saltlike silicides are formed with
Binary_compounds_of_silicon
Chemical compound
molecular binary metal oxide, the other examples being RuO4, OsO4, and the unstable Mn2O7. It adopts a centrosymmetric corner-shared bi-tetrahedral structure
Technetium(VII)_oxide
Natural number
a hexagonal number, and pentadecagonal number. It is also a centered tetrahedral number. 15 is a lucky number. 15 is the smallest positive number that
15_(number)
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 compound
compounds silane and methane, germane is tetrahedral. It burns in air to produce GeO2 and water. Germane is a group 14 hydride. Germane has been detected
Germane
3D symmetry group
S^{3}} , but which is not homotopic to it. Binary icosahedral group Icosian calculus Octahedral symmetry Tetrahedral symmetry Sir William Rowan Hamilton (1856)
Icosahedral_symmetry
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
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
Arrangement of spheres within a space
the hexagonal arrangement with a density of π/√27 ≈ 0.6046 and the tetrahedral arrangement with a density of π√3/16 ≈ 0.3401. Packings where all spheres
Sphere_packing
Natural number
since 56 = 28 + 14 + 8 + 4 + 2 {\displaystyle 56=28+14+8+4+2} . 56 is a tetrahedral number, since 56 = ( 8 3 ) {\displaystyle 56={\binom {8}{3}}} . It is
56_(number)
Integer whose multiples are digit rotations
787, 797, 809, 821, 857, 881, 929, 953, 967, 977, 991, ... For b = 2 (binary), these ps are (sequence A001122 in the OEIS) 3, 5, 11, 13, 19, 29, 37,
Cyclic_number
Crystallographic system where the unit cell is in the shape of a cube
corresponding tesselation, when lattice points are taken as vertices, is the tetrahedral-octahedral honeycomb. The face-centered cubic lattice is closely related
Cubic_crystal_system
Particular class of intermetallic phases
packing (TCP). Based on the tetrahedral units, FK crystallographic structures are classified into low and high polyhedral groups denoted by their coordination
Frank–Kasper_phases
that binary compounds with covalent character that have an average of 4 electrons per atom will have structures where both atoms are tetrahedrally coordinated
Grimm–Sommerfeld_rule
Group of elements in the periodic table
which shows at least 11 species, depending on pH and concentration. The tetrahedral orthovanadate ion, VO3− 4, is the principal species present at pH 12–14
Group_5_element
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
Type of chemical compound
which shows at least 11 species, depending on pH and concentration. The tetrahedral orthovanadate ion, VO3− 4, is the principal species present at pH 12–14
Vanadium_compounds
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
Natural number
triangular numbers (1, 3, 6, 10, 15, 21, 28), which makes it the seventh tetrahedral number. The number of divisors of 84 is 12. As no smaller number has
84_(number)
Prime such that p^2 divides 2^(p-1)-1
greater than a number with a periodic binary expansion, but up to a "bit pseudo-length" of 3500 of the tested binary numbers generated by combination of
Wieferich_prime
Two raised to an integer power
computers use a binary (base two) representation of numbers, a number system in which each place represents a power of 2, and the only possible binary digits ("bits")
Power_of_two
Polyatomic cation
N−H bonds are equivalent, being polar covalent bonds. The ion has a tetrahedral structure and is isoelectronic with methane and the borohydride anion
Ammonium
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
Natural number
A005894 (Centered tetrahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "Sloane's A000292 : Tetrahedral numbers". The On-Line
1000_(number)
This description is not unique to SiC, but also applies to other binary tetrahedral materials, such as zinc oxide and cadmium sulfide. A shorthand has
Polymorphs_of_silicon_carbide
coordination would be linear. Since the adduct compounds are not tetrahedral, the azido group can only act as a monodentate donor with two sterically active
Transition metal azide complex
Transition_metal_azide_complex
Compound of nitrogen with a formal oxidation state of –3
indium adopt the hexagonal wurtzite structure in which each atom occupies tetrahedral sites. For example, in aluminium nitride, each aluminium atom has four
Nitride
Natural number
standard for Unicode 10648 = 223, the smallest 5-digit cube 10660 = tetrahedral number 10671 = tetranacci number 10709 = highly cototient number 10744
10,000
Chemical compound
nature as the dark indigo blue mineral covellite. It is one of a number of binary compounds of copper and sulfur (see copper sulfide for an overview of this
Copper_monosulfide
Area of a right triangle with rational-numbered sides
rational number q is a congruent number depends only on its residue in the group Q ∗ / Q ∗ 2 , {\displaystyle \mathbb {Q} ^{*}/\mathbb {Q} ^{*2},} where
Congruent_number
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
Union of crystal groups with related structures and lattices
atom is tetrahedrally coordinated. The structure can also be described as an HCP lattice of zinc with sulfur atoms occupying half of the tetrahedral voids
Hexagonal_crystal_family
Natural number
number. 2017 is a sexy prime with 2011. 2024 = 23 × 11 × 23. It is a tetrahedral number. 2025 = 452, square of the sum of the first nine positive integers
2000_(number)
Ancient Mesopotamian board game
rolls of 4 (the royal game discovered by Woolley was found with 3 binary tetrahedral dice, which when rolled together can only produce 4 different results)
Royal_Game_of_Ur
Arithmetic operation
possible values for an n-bit integer binary number; for example, a byte may take 28 = 256 different values. The binary number system expresses any number
Exponentiation
Natural number
well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is the smallest number to appear six times in Pascal's triangle
120_(number)
Number of ways to pair up n objects
sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe
Telephone number (mathematics)
Telephone_number_(mathematics)
Natural number
different ways of partitioning 64 = 82 into a sum of square numbers. It is a tetrahedral number, the sum of the first ten triangular numbers, and a dodecahedral
220_(number)
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
molecule, because that would give C-C-C bond angles much greater than the tetrahedral angle of around 109.5°. Instead it is puckered and three-dimensional
Cycloheptane
Chemical element with atomic number 77 (Ir)
compound, each of the iridium atoms is bonded to the other three, forming a tetrahedral cluster. The discovery of Vaska's complex (IrCl(CO)[P(C6H5)3]2) opened
Iridium
Integer sequence in number theory
{\displaystyle 3^{4}7^{2}11^{2}19^{2}(-127)^{1}} . Subsequent work by a group at Brigham Young University found more examples similar to Voight's example
Descartes_number
Chemical compounds containing chromium
solutions of chrome alum and other water-soluble chromium(III) salts. A tetrahedral coordination of chromium(III) has been reported for the Cr-centered Keggin
Chromium_compounds
Chemical compound
Be(CN)2(py)2. Beryllium cyanide adopts a cubic crystal structure, with tetrahedrally coordinated beryllium by four cyanide anions. The tetrahedra are linked
Beryllium_cyanide
Chemical compound
cation is surrounded by eight F− anions in the form of a cube, which is tetrahedrally surrounded by four Yb2+. Greis, Ortwin; Haschke, John M. (1982). "Chapter
Ytterbium(II)_fluoride
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
Class of chemical compounds
not their structure. All known molecular phosphorus sulfides contain a tetrahedral array of four phosphorus atoms. P4S2 is also known but is unstable above
Phosphorus_sulfides
Number in combinatorics
each pair of parentheses surrounding two or more symbols or parenthesized groups, without any parentheses surrounding the entire sequence, and where "no
Schröder–Hipparchus_number
Chemical compounds containing copper
processes. As with other elements, the simplest compounds of copper are binary compounds, i.e. those containing only two elements, the principal examples
Copper_compounds
Chemical compounds containing iodine
quickest. Many periodates are known, including not only the expected tetrahedral IO− 4, but also square-pyramidal IO3− 5, octahedral orthoperiodate IO5−
Iodine_compounds
Number that is the sum of permutations of sub-samples of their own digits
332,639,667,360 is an Osiris number. Digit sum Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 138
Digit-reassembly_number
Chemical compounds containing iron
aqueous solution for some common iron ions are given below: The red-purple tetrahedral ferrate(VI) anion is such a strong oxidizing agent that it oxidizes nitrogen
Iron_compounds
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BINARY TETRAHEDRAL-GROUP
BINARY TETRAHEDRAL-GROUP
Girl/Female
Indian
(the wife of Sage Kashyap)
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.
Girl/Female
Indian
Modesty
Boy/Male
Latin
Happy; Cheerful.
Boy/Male
Indian
An intimate particle of the God of heaven
Girl/Female
English
Originally a diminutive used for names ending in -bina, like Albina, Columbina, and Robina, now...
Girl/Female
Hindu
Shore, Musical instrument, Goddess of wealth
Girl/Female
Hindu
Shore, Musical instrument, Goddess of wealth
Boy/Male
Indian, Punjabi, Sikh
Blessing
Male
Hindi/Indian
Variant spelling of Hindi Vijay, BIJAY means "victory."
Boy/Male
Irish
An ancient Irish name whos meaning is lost in antiquety.
Male
Scandinavian
Scandinavian form of Old Norse Einarr, EINAR means "lone warrior."
Male
English
English unisex form of Latin Hilarius and Hilaria, HILARY means "joyful; happy."Â Originally, this was strictly a masculine name.
Surname or Lastname
English
English : variant spelling of Vickery.
Female
Hebrew
Variant spelling of Hebrew Bina, BINAH means "intelligence, wisdom."Â
Boy/Male
American, Australian, French, German, Greek, Latin, Polish, Swedish
Cheerful; Happy; Joyful; Similar to Hilary
Male
Hindi/Indian
(विनय) Hindi name VINAY means "leading asunder."
Female
English
English pet form of German Belinda, possibly BINDY means "bright serpent" or "bright linden tree."
Female
Hebrew
(×‘Ö¼Ö´×™× Ö¸×”) Hebrew name BINA means "intelligence, wisdom."Â
Female
Turkish
Turkish name PINAR means "spring."
BINARY TETRAHEDRAL-GROUP
BINARY TETRAHEDRAL-GROUP
BINARY TETRAHEDRAL-GROUP
BINARY TETRAHEDRAL-GROUP
BINARY TETRAHEDRAL-GROUP
BINARY TETRAHEDRAL-GROUP
BINARY TETRAHEDRAL-GROUP
adv.
In a tetrahedral manner.
n.
Wine made in the Canary Islands; sack.
v. i.
To perform the canary dance; to move nimbly; to caper.
a.
Pertaining or related to a tetrahedron, or to the system of hemihedral forms to which the tetrahedron belongs.
a.
Having, or composed of, four sides.
n.
A binary compound of silicon, or one regarded as binary.
a.
Of or pertaining to the Canary Islands; as, canary wine; canary birds.
a.
Containing ten; tenfold; proceeding by tens; as, the denary, or decimal, scale.
n.
A canary bird.
a.
Of or pertaining to the urine; as, the urinary bladder; urinary excretions.
a.
lasting for one day; as, a diary fever.
a.
Relating or belonging to bile; conveying bile; as, biliary acids; biliary ducts.
n.
A binary compound of zinc.
n.
See Finery.
n.
A binary compound of phosphorus.
n.
A register of daily events or transactions; a daily record; a journal; a blank book dated for the record of daily memoranda; as, a diary of the weather; a physician's diary.
n.
A pale yellow color, like that of a canary bird.
a.
Of a pale yellowish color; as, Canary stone.
a.
Having the form of the regular tetrahedron.
n.
A binary compound of selenium, or a compound regarded as binary; as, ethyl selenide.
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