symmetric group S5
E904572
Coxeter group
Weyl group
finite group
non‑abelian group
permutation group
simple group
symmetric group
The symmetric group S5 is the group of all permutations of five elements, a fundamental finite group of order 120 that plays a key role in group theory and Galois theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| symmetric group S5 canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11098891 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: symmetric group S5 Context triple: [Clebsch diagonal surface, hasAutomorphismGroup, symmetric group S5]
-
A.
Burnside's lemma
Burnside's lemma is a result in group theory and combinatorics that counts distinct configurations under symmetries by averaging the number of fixed points of group actions.
-
B.
PSL(2,7)
PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
-
C.
Harada–Norton group
The Harada–Norton group is one of the 26 sporadic simple groups in finite group theory, notable for its large order and close relationship to the Monster group.
-
D.
Galois group
A Galois group is the group of field automorphisms of a field extension that captures the symmetries of its algebraic equations and underpins much of modern algebra and number theory.
-
E.
Carian-Sidetic-Pisidian group
The Carian-Sidetic-Pisidian group is a subgroup of the Luwic branch of the Anatolian Indo-European languages, encompassing the closely related ancient languages Carian, Sidetic, and Pisidian once spoken in southwestern Anatolia.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: symmetric group S5 Target entity description: The symmetric group S5 is the group of all permutations of five elements, a fundamental finite group of order 120 that plays a key role in group theory and Galois theory.
-
A.
Burnside's lemma
Burnside's lemma is a result in group theory and combinatorics that counts distinct configurations under symmetries by averaging the number of fixed points of group actions.
-
B.
PSL(2,7)
PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
-
C.
Harada–Norton group
The Harada–Norton group is one of the 26 sporadic simple groups in finite group theory, notable for its large order and close relationship to the Monster group.
-
D.
Galois group
A Galois group is the group of field automorphisms of a field extension that captures the symmetries of its algebraic equations and underpins much of modern algebra and number theory.
-
E.
Carian-Sidetic-Pisidian group
The Carian-Sidetic-Pisidian group is a subgroup of the Luwic branch of the Anatolian Indo-European languages, encompassing the closely related ancient languages Carian, Sidetic, and Pisidian once spoken in southwestern Anatolia.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Coxeter group
ⓘ
Weyl group ⓘ finite group ⓘ non‑abelian group ⓘ permutation group ⓘ simple group ⓘ symmetric group ⓘ |
| definedAs | group of all permutations of a 5‑element set ⓘ |
| hasAutomorphismGroup | symmetric group S5 NERFINISHED ⓘ |
| hasCenter | trivial group ⓘ |
| hasCompositionFactor | alternating group A5 GENERATED ⓘ |
| hasCompositionLength | 2 GENERATED ⓘ |
| hasConjugacyClassCorrespondence | partitions of 5 GENERATED ⓘ |
| hasCoxeterPresentation | generated by s1,s2,s3,s4 with (si)^2=1,(si sj)^3=1 for |i−j|=1,(si sj)^2=1 for |i−j|>1 ⓘ |
| hasCoxeterType | A4 GENERATED ⓘ |
| hasDegree | 5 ⓘ |
| hasDerivedSubgroup | alternating group A5 NERFINISHED ⓘ |
| hasElementOfOrder |
10
ⓘ
12 ⓘ 2 ⓘ 3 ⓘ 4 ⓘ 5 ⓘ 6 ⓘ |
| hasExponent | 60 ⓘ |
| hasIndex | 2 subgroup alternating group A5 ⓘ |
| hasMinimalNumberOfGenerators | 2 ⓘ |
| hasNaturalAction | on a 5‑element set ⓘ |
| hasNormalSubgroup | alternating group A5 NERFINISHED ⓘ |
| hasNumberOfConjugacyClasses | 7 ⓘ |
| hasOrder | 120 ⓘ |
| hasRoleIn |
Galois theory
ⓘ
classification of finite simple groups ⓘ |
| hasSignHomomorphism | onto cyclic group of order 2 ⓘ |
| hasSylow2SubgroupOrder | 8 ⓘ |
| hasSylow3SubgroupOrder | 3 ⓘ |
| hasSylow5SubgroupOrder | 5 ⓘ |
| hasTransitiveAction | on 5 points ⓘ |
| hasTrivialCenter | true ⓘ |
| isCompleteGroup | true ⓘ |
| isFinite | true ⓘ |
| isGeneratedBy |
a transposition and a 5‑cycle
ⓘ
adjacent transpositions (1 2),(2 3),(3 4),(4 5) ⓘ |
| isNonAbelian | true ⓘ |
| isPrimitivePermutationGroup | true ⓘ |
| isSimple | true ⓘ |
| isSmallestSymmetricGroupWithNonSolvableGaloisGroup | true ⓘ |
| isSmallestSymmetricGroupWithNonSolvableSubgroup | true ⓘ |
| isSolvable | false ⓘ |
| quotientByA5 | cyclic group of order 2 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: symmetric group S5 Description of subject: The symmetric group S5 is the group of all permutations of five elements, a fundamental finite group of order 120 that plays a key role in group theory and Galois theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Clebsch diagonal surface