Properties

Label 6T15
6T15 1 2 1->2 1->2 3 2->3 3->1 4 3->4 5 4->5 6 5->6 6->3
Degree $6$
Order $360$
Cyclic no
Abelian no
Solvable no
Transitivity $4$
Primitive yes
$p$-group no
Group: $A_6$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(6, 15);
 
Copy content sage:G = TransitiveGroup(6, 15)
 
Copy content oscar:G = transitive_group(6, 15)
 
Copy content gap:G := TransitiveGroup(6, 15);
 

Group invariants

Abstract group:  $A_6$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  no
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $6$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $15$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
CHM label:   $A6$
Parity:  $1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  4
Primitive:  yes
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(6).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(6), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(6), G));
 
Generators:  $(1,2,3)$, $(1,2)(3,4,5,6)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Low degree siblings

6T15, 10T26, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 40T304, 45T49

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{6}$ $1$ $1$ $0$ $()$
2A $2^{2},1^{2}$ $45$ $2$ $2$ $(1,4)(3,5)$
3A $3,1^{3}$ $40$ $3$ $2$ $(2,5,4)$
3B $3^{2}$ $40$ $3$ $4$ $(1,6,2)(3,4,5)$
4A $4,2$ $90$ $4$ $4$ $(1,5,4,3)(2,6)$
5A1 $5,1$ $72$ $5$ $4$ $(1,4,6,2,3)$
5A2 $5,1$ $72$ $5$ $4$ $(1,6,3,4,2)$

Malle's constant $a(G)$:     $1/2$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 3A 3B 4A 5A1 5A2
Size 1 45 40 40 90 72 72
2 P 1A 1A 3A 3B 2A 5A2 5A1
3 P 1A 2A 1A 1A 4A 5A2 5A1
5 P 1A 2A 3A 3B 4A 1A 1A
Type
360.118.1a R 1 1 1 1 1 1 1
360.118.5a R 5 1 −1 2 −1 0 0
360.118.5b R 5 1 2 −1 −1 0 0
360.118.8a1 R 8 0 −1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52
360.118.8a2 R 8 0 −1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5
360.118.9a R 9 1 0 0 1 −1 −1
360.118.10a R 10 −2 1 1 0 0 0

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{6} + 3 x^{5} + 15 x^{4} + t x^{3} + 12 x - 4$ Copy content Toggle raw display