About: Binary tetrahedral group

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In 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 (2,3,3) of order 12 by a cyclic group of order 2, and is the preimage 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. Shephard or 3[3]3 and by Coxeter, is isomorphic to the binary tetrahedral group.

Attributes	Values
rdf:type	yago:WikicatBinaryPolyhedralGroups yago:Abstraction100002137 yago:Group100031264
rdfs:label	Binary tetrahedral group (en) Бинарная группа тетраэдра (ru)
rdfs:comment	In 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 (2,3,3) of order 12 by a cyclic group of order 2, and is the preimage 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. Shephard or 3[3]3 and by Coxeter, is isomorphic to the binary tetrahedral group. (en) В математике бинарная группа тетраэдра (обозначается как 2T или <2,3,3>) — это некоторая неабелева группа 24-го порядка. Группа является расширением тетраэдральной группы T (или (2,3,3)) 12-го порядка циклической группы 2-го порядка и является прообразом группы тетраэдра для 2:1 специальной ортогональной группы спинорной группой. Отсюда следует, чтобинарная группа тетраэдра — дискретная подгруппа группы Spin(3) 24-го порядка. (ru)
foaf:depiction
dcterms:subject	Binary polyhedral groups
Wikipage page ID	10997586 (xsd:integer)
Wikipage revision ID	1117496097 (xsd:integer)
Link from a Wikipage to another Wikipage	Projective special linear group Quaternions and spatial rotation Paul Frampton Unit quaternion Complex reflection group Covering groups of the alternating and symmetric groups Coxeter Mathematics Geoffrey Colin Shephard Normal subgroup Yang–Mills theory Convex hull Short exact sequence Simplex Spin group Superperfect group 24-cell Alternating group Cyclic group Finite field Conjugation (group theory) Quaternion group Ring (mathematics) Binary cyclic group Binary dihedral group Binary icosahedral group Binary octahedral group Binary polyhedral group Binary polyhedral groups Automorphism group Special orthogonal group Group extension Group of units Order (group theory) Semidirect product Special linear group Nonabelian group Chen Ning Yang Convex regular 4-polytope Preimage Split exact sequence Universal perfect central extension Lipschitz unit Tetrahedral group Center of a group Covering homomorphism Discrete subgroup Group presentation Hurwitz integer Index (group theory) Inner automorphism group Sp(1)
sameAs	Binary tetrahedral group Binary tetrahedral group Binary tetrahedral group Binary tetrahedral group Binary tetrahedral group
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has abstract	In 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 (2,3,3) of order 12 by a cyclic group of order 2, and is the preimage 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. Shephard or 3[3]3 and by Coxeter, is isomorphic to the binary tetrahedral group. The binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin(3) ≅ Sp(1), where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.) (en) В математике бинарная группа тетраэдра (обозначается как 2T или <2,3,3>) — это некоторая неабелева группа 24-го порядка. Группа является расширением тетраэдральной группы T (или (2,3,3)) 12-го порядка циклической группы 2-го порядка и является прообразом группы тетраэдра для 2:1 специальной ортогональной группы спинорной группой. Отсюда следует, чтобинарная группа тетраэдра — дискретная подгруппа группы Spin(3) 24-го порядка. Бинарную группу тетраэдра проще всего описать как дискретную подгруппу единиц кватернионов при изоморфизме , где — мультипликативная группа единиц кватернионов (см. описание этого гомоморфизма в статье кватернионы и вращение пространства). (ru)
prov:wasDerivedFrom	wikipedia-en:Binary_tetrahedral_group?oldid=1117496097&ns=0
page length (characters) of wiki page	9919 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Binary_tetrahedral_group
is Link from a Wikipage to another Wikipage of	Hurwitz quaternion Paul Frampton Complex polytope Complex reflection group Covering groups of the alternating and symmetric groups Quaternionic polytope Near-field (mathematics)

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