S5
E904571
S5 is the symmetric group on five elements, a fundamental non-abelian finite group that plays a key role in permutation group theory and Galois theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| S5 canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11098890 — 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: S5 Context triple: [Clebsch diagonal surface, hasSymmetryGroup, S5]
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A.
S5
S5 is a line of the Berlin S-Bahn rapid transit network serving routes between central Berlin and its eastern suburbs.
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B.
S5
S5 is a commuter rail line of the Stuttgart S-Bahn network serving the Stuttgart metropolitan area in Germany.
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C.
S5
S5 is a regional S-Bahn rail line within Germany’s Rhine-Ruhr metropolitan transit network, connecting key cities and suburbs in the area.
-
D.
S51
S51 is a New York City bus route on Staten Island that connects the Shore Acres area with other parts of the borough.
-
E.
S52
S52 is a bus service route that provides public transportation access to the village of Grasmere.
- 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: S5 Target entity description: S5 is the symmetric group on five elements, a fundamental non-abelian finite group that plays a key role in permutation group theory and Galois theory.
-
A.
S5
S5 is a line of the Berlin S-Bahn rapid transit network serving routes between central Berlin and its eastern suburbs.
-
B.
S5
S5 is a commuter rail line of the Stuttgart S-Bahn network serving the Stuttgart metropolitan area in Germany.
-
C.
S5
S5 is a regional S-Bahn rail line within Germany’s Rhine-Ruhr metropolitan transit network, connecting key cities and suburbs in the area.
-
D.
S51
S51 is a New York City bus route on Staten Island that connects the Shore Acres area with other parts of the borough.
-
E.
S52
S52 is a bus service route that provides public transportation access to the village of Grasmere.
- F. None of above. chosen
Statements (52)
| Predicate | Object |
|---|---|
| instanceOf |
abstract algebraic structure
ⓘ
finite group ⓘ non-abelian group ⓘ permutation group ⓘ symmetric group ⓘ |
| actsOn | set of five elements ⓘ |
| appearsInGaloisTheoryAs | Galois group of many quintic extensions of Q ⓘ |
| containsSubgroup |
A5
ⓘ
C2 ⓘ C5 ⓘ D5 ⓘ S4 ⓘ V4 ⓘ |
| hasAutomorphismGroupIsomorphicTo | S5 ⓘ |
| hasCayleyGraphUsedIn | combinatorics and group theory ⓘ |
| hasCenter | trivial group ⓘ |
| hasConjugacyClass |
2-cycles
GENERATED
ⓘ
3-cycles GENERATED ⓘ 4-cycles GENERATED ⓘ 5-cycles GENERATED ⓘ identity permutation GENERATED ⓘ products of a 3-cycle and a disjoint 2-cycle GENERATED ⓘ products of two disjoint 2-cycles GENERATED ⓘ |
| hasExponent | 60 ⓘ |
| hasIndex | 2 in S5 over A5 ⓘ |
| hasNormalSubgroup | A5 NERFINISHED ⓘ |
| hasNumberOfConjugacyClasses | 7 ⓘ |
| hasOrder | 120 ⓘ |
| hasOrderOfSylow2Subgroup | 8 ⓘ |
| hasOrderOfSylow3Subgroup | 3 ⓘ |
| hasOrderOfSylow5Subgroup | 5 ⓘ |
| hasOuterAutomorphismGroup | trivial group ⓘ |
| hasQuotientIsomorphicTo | C2 ⓘ |
| hasSignHomomorphismTo | C2 ⓘ |
| hasTrivialCenter | true ⓘ |
| isCompleteGroup | true ⓘ |
| isDenotedBy |
Sym(5)
ⓘ
Σ5 ⓘ |
| isDoublyTransitive | true ⓘ |
| isGaloisGroupOf | generic irreducible quintic polynomial over Q ⓘ |
| isGeneratedBy |
(1 2) and (1 2 3 4 5)
ⓘ
a 2-cycle and a 5-cycle ⓘ |
| isGeneratedByTranspositions | true ⓘ |
| isGroupOfPermutationsOf | five-element set ⓘ |
| isIsomorphicTo | group of all bijections on a 5-element set ⓘ |
| isNonAbelian | true ⓘ |
| isNonAbelianFor | n ≥ 3 ⓘ |
| isPrimitivePermutationGroup | true ⓘ |
| isSimple | true ⓘ |
| isTransitiveGroupOn | 5 points ⓘ |
| kernelOfSignHomomorphism | A5 ⓘ |
| quotientBy | A5 ⓘ |
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: S5 Description of subject: S5 is the symmetric group on five elements, a fundamental non-abelian finite group that plays a key role in permutation 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