Properties

Label 6T1
6T1 1 2 1->2 3 2->3 4 3->4 5 4->5 6 5->6 6->1
Degree $6$
Order $6$
Cyclic yes
Abelian yes
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_6$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(6, 1);
 
Copy content sage:G = TransitiveGroup(6, 1)
 
Copy content oscar:G = transitive_group(6, 1)
 
Copy content gap:G := TransitiveGroup(6, 1);
 

Group invariants

Abstract group:  $C_6$
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:  $6=2 \cdot 3$
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:  yes
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:  yes
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:  yes
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:  $1$
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 $t$:  $1$
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:   $C(6) = 6 = 3[x]2$
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:  1
Primitive:  no
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);
 
$\card{\Aut(F/K)}$:  $6$
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:  $(1,2,3,4,5,6)$
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

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Low degree siblings

There are no siblings 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 $2^{3}$ $1$ $2$ $3$ $(1,4)(2,5)(3,6)$
3A1 $3^{2}$ $1$ $3$ $4$ $(1,3,5)(2,4,6)$
3A-1 $3^{2}$ $1$ $3$ $4$ $(1,5,3)(2,6,4)$
6A1 $6$ $1$ $6$ $5$ $(1,2,3,4,5,6)$
6A-1 $6$ $1$ $6$ $5$ $(1,6,5,4,3,2)$

Malle's constant $a(G)$:     $1/3$

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 3A1 3A-1 6A1 6A-1
Size 1 1 1 1 1 1
2 P 1A 1A 3A-1 3A1 3A1 3A-1
3 P 1A 2A 1A 1A 2A 2A
Type
6.2.1a R 1 1 1 1 1 1
6.2.1b R 1 −1 1 1 −1 −1
6.2.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1
6.2.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3
6.2.1d1 C 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1
6.2.1d2 C 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3

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

$f_{ 1 } =$ $x^{6} + 2 t x^{5} + \left(-5 t - 15\right) x^{4} + 20 x^{3} + 5 t x^{2} + \left(-2 t - 6\right) x + 1$ Copy content Toggle raw display