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