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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.