PGL(2,7)
E904567
PGL(2,7) is the projective general linear group of 2×2 invertible matrices over the finite field with 7 elements, a finite group of order 336 that acts as the full collineation group of the projective line over that field.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
finite group
ⓘ
linear group ⓘ matrix group ⓘ permutation group ⓘ projective general linear group ⓘ |
| actsFaithfullyOn | projective line over F_7 ⓘ |
| actsOn | projective line over F_7 ⓘ |
| actsSharply3TransitivelyOn | projective line over F_7 ⓘ |
| actsTransitivelyOn | projective line over F_7 ⓘ |
| constructedAs | GL(2,7)/Z(GL(2,7)) NERFINISHED ⓘ |
| containsSubgroupIsomorphicTo | PSL(2,7) NERFINISHED ⓘ |
| definedOver | finite field F_7 ⓘ |
| degreeOfNaturalPermutationRepresentation | 8 ⓘ |
| embedsInto | S_8 ⓘ |
| hasAbelianization | C2 ⓘ |
| hasCenter | trivial group ⓘ |
| hasConjugacyClasses | 9 ⓘ |
| hasDerivedSubgroup | PSL(2,7) NERFINISHED ⓘ |
| hasElementOfOrder |
12
ⓘ
14 ⓘ 2 ⓘ 21 ⓘ 3 ⓘ 4 ⓘ 6 ⓘ 7 ⓘ 8 ⓘ |
| hasExponent | 168 ⓘ |
| hasIndex | 2 subgroup PSL(2,7) ⓘ |
| hasMinimalNormalSubgroup | PSL(2,7) NERFINISHED ⓘ |
| hasNormalSubgroup | PSL(2,7) NERFINISHED ⓘ |
| hasOrder | 336 ⓘ |
| hasOrderFactorization | 2^4·3·7 ⓘ |
| hasOrderOfSylow2Subgroup | 16 ⓘ |
| hasOrderOfSylow3Subgroup | 3 ⓘ |
| hasOrderOfSylow7Subgroup | 7 ⓘ |
| hasOuterAutomorphisms | no ⓘ |
| hasSimpleNormalSubgroup | PSL(2,7) NERFINISHED ⓘ |
| hasSmallGroupIdentifier | SmallGroup(336, 208) ⓘ |
| isAutomorphismGroupOf | Fano plane NERFINISHED ⓘ |
| isCollineationGroupOf | projective line over F_7 ⓘ |
| isExtensionOf | PSL(2,7) by C2 NERFINISHED ⓘ |
| isNotSimple | true ⓘ |
| isomorphicTo |
PGL(2,7)
NERFINISHED
ⓘ
PΓL(2,7) NERFINISHED ⓘ |
| isPerfect | false ⓘ |
| isPrimitivePermutationGroup | true ⓘ |
| isSubgroupOf | S_8 ⓘ |
| isTransitiveSubgroupOf | S_8 ⓘ |
| quotientOf | GL(2,7) NERFINISHED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.