SL(2,7)
E904566
SL(2,7) is the special linear group of 2×2 matrices with determinant 1 over the finite field with 7 elements, a non-abelian finite group of order 336 that plays an important role in group theory and geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| SL(2,7) canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11098582 — 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: SL(2,7) Context triple: [PSL(2,7), isQuotientOf, SL(2,7)]
-
A.
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.
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B.
SL(2,ℤ)
SL(2,ℤ) is the group of 2×2 integer matrices with determinant 1, fundamental in number theory, geometry, and the theory of modular forms.
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C.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
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D.
SL(2,R)
SL(2,R) is the Lie group of 2×2 real matrices with determinant 1, fundamental in representation theory, geometry, and the study of symmetries in mathematics and physics.
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E.
special linear group SL(n,C)
The special linear group SL(n,ℂ) is the Lie group of n×n complex matrices with determinant 1, fundamental in representation theory, geometry, and many areas of modern mathematics and physics.
- 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: SL(2,7) Target entity description: SL(2,7) is the special linear group of 2×2 matrices with determinant 1 over the finite field with 7 elements, a non-abelian finite group of order 336 that plays an important role in group theory and geometry.
-
A.
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.
-
B.
SL(2,ℤ)
SL(2,ℤ) is the group of 2×2 integer matrices with determinant 1, fundamental in number theory, geometry, and the theory of modular forms.
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C.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
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D.
SL(2,R)
SL(2,R) is the Lie group of 2×2 real matrices with determinant 1, fundamental in representation theory, geometry, and the study of symmetries in mathematics and physics.
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E.
special linear group SL(n,C)
The special linear group SL(n,ℂ) is the Lie group of n×n complex matrices with determinant 1, fundamental in representation theory, geometry, and many areas of modern mathematics and physics.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
finite group
ⓘ
linear group ⓘ matrix group ⓘ non-abelian group ⓘ perfect group ⓘ special linear group ⓘ |
| abelianizationIsTrivial | true ⓘ |
| actsNaturallyOn | 2-dimensional vector space over F7 ⓘ |
| appearsIn |
Klein quartic automorphism group theory
ⓘ
theory of Hurwitz groups ⓘ |
| centerIsIsomorphicTo | C2 ⓘ |
| definedOver | finite field F7 ⓘ |
| determinantMapKernelIs | SL(2,7) NERFINISHED ⓘ |
| hasCenter | {±I} ⓘ |
| hasCenterOrder | 2 ⓘ |
| hasDeterminantCondition | determinant 1 ⓘ |
| hasDeterminantMapTo | F7* ⓘ |
| hasElementOfOrder |
12
ⓘ
14 ⓘ 2 ⓘ 21 ⓘ 24 ⓘ 3 ⓘ 4 ⓘ 6 ⓘ 7 ⓘ 8 ⓘ |
| hasExponent | 84 ⓘ |
| hasMatrixSize | 2×2 ⓘ |
| hasOrder | 336 ⓘ |
| hasQuotientOrder | 168 ⓘ |
| hasSchurMultiplier | C2 ⓘ |
| hasSimpleQuotient | PSL(2,7) NERFINISHED ⓘ |
| hasSmallestFieldSize | 7 for nontrivial SL(2,q) with q>3 ⓘ |
| hasSylowSubgroupForPrime |
2
ⓘ
3 ⓘ 7 ⓘ |
| is2FoldCoverOf | PSL(2,7) NERFINISHED ⓘ |
| isDoubleCoverOf | PSL(2,7) NERFINISHED ⓘ |
| isIsomorphicTo | 2·PSL(2,7) NERFINISHED ⓘ |
| isNonAbelian | true ⓘ |
| isNonSolvable | true ⓘ |
| isPerfect | true ⓘ |
| isSubgroupOf | GL(2,7) NERFINISHED ⓘ |
| isUniversalCentralExtensionOf | PSL(2,7) NERFINISHED ⓘ |
| orderFactorization | 2^4·3·7 ⓘ |
| quotientByCenterIs | PSL(2,7) NERFINISHED ⓘ |
| usedIn |
finite geometry
ⓘ
finite group theory ⓘ representation theory ⓘ |
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: SL(2,7) Description of subject: SL(2,7) is the special linear group of 2×2 matrices with determinant 1 over the finite field with 7 elements, a non-abelian finite group of order 336 that plays an important role in group theory and geometry.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.