Symmetric group-3AS3
type: Ontology
creation date: 2026-06-05T04:29:41+00:00
imports: http://semantic-mediawiki.org/swivt/1.0

Symmetric group:S3
type: Subject
type: Category:Particular groups
type: Category:General affine groups
type: Category:Symmetric groups
type: Category:Dihedral groups
page: https://groupprops.subwiki.org/wiki/Symmetric_group-3AS3
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
dissatisfies property: Nilpotent group
dissatisfies property: 2-Engel group
importance rank: 1
member of family: Coxeter group
member of family: Symmetric group
member of family: Symmetric group on finite set
member of family: Symmetric group of prime degree
member of family: Symmetric group of prime power degree
member of family: Dihedral group
member of family: Special linear group
member of family: Special linear group of degree two
member of family: Field:F2
member of family: Projective special linear group
member of family: Projective special linear group of degree two
member of family: General linear group of degree two
member of family: Projective general linear group
member of family: Projective general linear group of degree two
member of family: General affine group
member of family: General semilinear group
member of family: General semilinear group of degree one
member of family: Von Dyck group
order: 6
page class: Term
satisfies property: Metacyclic group
satisfies property: Supersolvable group
satisfies property: Solvable group
satisfies property: Rational-representation group
satisfies property: Rational group
satisfies property: Ambivalent group
satisfies property: T-group
satisfies property: Schur-trivial group
satisfies property: Finite group
satisfies property: Monolithic group
satisfies property: One-headed group
satisfies property: SC-group
satisfies property: Complete group
satisfies property: Group isomorphic to its automorphism group
satisfies property: Frobenius group
satisfies property: Camina group
satisfies property: Z-group
wiki page modification date: 2026-01-10T01:48:03Z
modification date#aux: 2461050.5750347
wiki page sort key: Symmetric group:S3

http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_dc121d11b0504b92b35bb940e4cb7913
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number: 6
page: http://groupprops.subwiki.org/wiki/Special:URIResolver/Order_of_a_group

http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_fe9f576d93fd57ea07316b4747cb84f1
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page: http://groupprops.subwiki.org/wiki/Special:URIResolver/Exponent_of_a_group

http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_e25716f2c9902d61b874a5e945bcd4e1
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http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_5e221dcc4544cb5197d6f466b985fadc
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http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_483f66b5c3b43a824e78c384de7e7fde
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http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_056ff4b247cc33ab04306871c1e72635
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http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_637a4127b1e354a9c40cb4ca4bca15f9
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http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_b27b517cda7fee96edbf052a50a8f056
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http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_f5a540bd8aa479f7d41e8c22a764dee7
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http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_QUERYc4d2cd79117c8c973d3ed51ec51e80e9
type: Subject
master page: Symmetric group:S3
wiki namespace: 0
query depth: 2
query format: table
query size: 10
query string: Number of equivalence classes under rational conjugacy 3 Order of a group 6
wiki page sort key: Symmetric group:S3# QUERYc4d2cd79117c8c973d3ed51ec51e80e9

http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_QUERY3e9549def5152797affe3249b0f1a6e8
type: Subject
master page: Symmetric group:S3
wiki namespace: 0
query depth: 2
query format: table
query size: 5
query string: Number of equivalence classes under rational conjugacy 3
wiki page sort key: Symmetric group:S3# QUERY3e9549def5152797affe3249b0f1a6e8

http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_QUERY6c5e593e7d1d675df81ea6201e6ecaf3
type: Subject
master page: Symmetric group:S3
wiki namespace: 0
query depth: 2
query format: table
query size: 10
query string: Number of conjugacy classes of rational elements 3 Order of a group 6
wiki page sort key: Symmetric group:S3# QUERY6c5e593e7d1d675df81ea6201e6ecaf3

http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_QUERY282fa30781dec06a76787b5534dabf9f
type: Subject
master page: Symmetric group:S3
wiki namespace: 0
query depth: 2
query format: table
query size: 5
query string: Number of conjugacy classes of rational elements 3
wiki page sort key: Symmetric group:S3# QUERY282fa30781dec06a76787b5534dabf9f

http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_QUERY84bab39de83f7433bb0367ce65d19451
type: Subject
master page: Symmetric group:S3
wiki namespace: 0
query depth: 2
query format: table
query size: 10
query string: Number of normal subgroups 3 Order of a group 6
wiki page sort key: Symmetric group:S3# QUERY84bab39de83f7433bb0367ce65d19451

http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_QUERYbff4421548198e5201962135e71005ed
type: Subject
master page: Symmetric group:S3
wiki namespace: 0
query depth: 2
query format: table
query size: 5
query string: Number of normal subgroups 3
wiki page sort key: Symmetric group:S3# QUERYbff4421548198e5201962135e71005ed

http://groupprops.subwiki.org/wiki/Special:URIResolver/Symmetric_group-3AS3-23_QUERY416654683b3f91b55b3d9d5d2b321070
type: Subject
master page: Symmetric group:S3
wiki namespace: 0
query depth: 0
query format: plainlist
query size: 1
query string: Groups of order 6
wiki page sort key: Symmetric group:S3# QUERY416654683b3f91b55b3d9d5d2b321070

Order-conjugate and Hall not implies order-dominating
type: Subject
page: https://groupprops.subwiki.org/wiki/Order-2Dconjugate_and_Hall_not_implies_order-2Ddominating
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: Order-conjugate and Hall not implies order-dominating

Nilpotent not implies nilpotent automorphism group
type: Subject
page: https://groupprops.subwiki.org/wiki/Nilpotent_not_implies_nilpotent_automorphism_group
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: Nilpotent not implies nilpotent automorphism group

CA not implies nilpotent
type: Subject
page: https://groupprops.subwiki.org/wiki/CA_not_implies_nilpotent
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: CA not implies nilpotent

Order-conjugate not implies order-dominating
type: Subject
page: https://groupprops.subwiki.org/wiki/Order-2Dconjugate_not_implies_order-2Ddominating
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: Order-conjugate not implies order-dominating

Normal-isomorph-free not implies isomorph-free in finite
type: Subject
page: https://groupprops.subwiki.org/wiki/Normal-2Disomorph-2Dfree_not_implies_isomorph-2Dfree_in_finite
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: Normal-isomorph-free not implies isomorph-free in finite

Supersolvable not implies nilpotent
type: Subject
page: https://groupprops.subwiki.org/wiki/Supersolvable_not_implies_nilpotent
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: Supersolvable not implies nilpotent

Characteristic not implies characteristic-isomorph-free in finite
type: Subject
page: https://groupprops.subwiki.org/wiki/Characteristic_not_implies_characteristic-2Disomorph-2Dfree_in_finite
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: Characteristic not implies characteristic-isomorph-free in finite

Class two not implies abelian automorphism group
type: Subject
page: https://groupprops.subwiki.org/wiki/Class_two_not_implies_abelian_automorphism_group
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: Class two not implies abelian automorphism group

Transitively normal not implies conjugacy-closed normal
type: Subject
page: https://groupprops.subwiki.org/wiki/Transitively_normal_not_implies_conjugacy-2Dclosed_normal
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: Transitively normal not implies conjugacy-closed normal

Finite not implies composition factor-permutable
type: Subject
page: https://groupprops.subwiki.org/wiki/Finite_not_implies_composition_factor-2Dpermutable
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: Finite not implies composition factor-permutable

Modular not implies permutable
type: Subject
page: https://groupprops.subwiki.org/wiki/Modular_not_implies_permutable
wiki namespace: 0
wiki page content language: en
particular example: Symmetric group:S3
wiki page sort key: Modular not implies permutable

Direct product of Z5 and S3
type: Subject
page: https://groupprops.subwiki.org/wiki/Direct_product_of_Z5_and_S3
wiki namespace: 0
wiki page content language: en
defining ingredient: Symmetric group:S3
wiki page sort key: Direct product of Z5 and S3

S3 in S5
type: Subject
page: https://groupprops.subwiki.org/wiki/S3_in_S5
wiki namespace: 0
wiki page content language: en
subgroup part: Symmetric group:S3
wiki page sort key: S3 in S5

Particular example
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3AParticular_example
wiki namespace: 132
wiki page content language: en
wiki page sort key: Particular example

Defining ingredient
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3ADefining_ingredient
wiki namespace: 132
wiki page content language: en
wiki page sort key: Defining ingredient

Subgroup part
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3ASubgroup_part
wiki namespace: 132
wiki page content language: en
wiki page sort key: Subgroup part
type: wpg