↑

Properties

Label 13T6
13T6 1 2 1->2 1->2 3 2->3 4 2->4 3->4 6 3->6 5 4->5 8 4->8 5->6 10 5->10 7 6->7 12 6->12 7->1 7->8 8->3 9 8->9 9->5 9->10 10->7 11 10->11 11->9 11->12 12->11 13 12->13 13->1
Degree $13$
Order $156$
Cyclic no
Abelian no
Solvable yes
Transitivity $2$
Primitive yes
$p$-group no
Group: $F_{13}$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(13, 6);
 
Copy content sage:G = TransitiveGroup(13, 6)
 
Copy content oscar:G = transitive_group(13, 6)
 
Copy content gap:G := TransitiveGroup(13, 6);
 

Group invariants

Abstract group:  $F_{13}$
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:  $156=2^{2} \cdot 3 \cdot 13$
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:  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:   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 $n$:  $13$
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 $t$:  $6$
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:   $F_{156}(13)=13:12$
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:  2
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(13).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(13), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(13), G));
 
Generators:  $(1,2,3,4,5,6,7,8,9,10,11,12,13)$, $(1,2,4,8,3,6,12,11,9,5,10,7)$
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)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

26T8, 39T11

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{13}$ $1$ $1$ $0$ $()$
2A $2^{6},1$ $13$ $2$ $6$ $( 1, 7)( 2, 6)( 3, 5)( 8,13)( 9,12)(10,11)$
3A1 $3^{4},1$ $13$ $3$ $8$ $( 1, 8, 3)( 2,11,12)( 5, 7,13)( 6,10, 9)$
3A-1 $3^{4},1$ $13$ $3$ $8$ $( 1, 3, 8)( 2,12,11)( 5,13, 7)( 6, 9,10)$
4A1 $4^{3},1$ $13$ $4$ $9$ $( 1, 2, 7, 6)( 3,12, 5, 9)( 8,11,13,10)$
4A-1 $4^{3},1$ $13$ $4$ $9$ $( 1, 6, 7, 2)( 3, 9, 5,12)( 8,10,13,11)$
6A1 $6^{2},1$ $13$ $6$ $10$ $( 1, 5, 8, 7, 3,13)( 2, 9,11, 6,12,10)$
6A-1 $6^{2},1$ $13$ $6$ $10$ $( 1,13, 3, 7, 8, 5)( 2,10,12, 6,11, 9)$
12A1 $12,1$ $13$ $12$ $11$ $( 1,10, 5, 2, 8, 9, 7,11, 3, 6,13,12)$
12A-1 $12,1$ $13$ $12$ $11$ $( 1,12,13, 6, 3,11, 7, 9, 8, 2, 5,10)$
12A5 $12,1$ $13$ $12$ $11$ $( 1, 9,13, 2, 3,10, 7,12, 8, 6, 5,11)$
12A-5 $12,1$ $13$ $12$ $11$ $( 1,11, 5, 6, 8,12, 7,10, 3, 2,13, 9)$
13A $13$ $12$ $13$ $12$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

Malle's constant $a(G)$:     $1/6$

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 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5 13A
Size 1 13 13 13 13 13 13 13 13 13 13 13 12
2 P 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 6A1 6A-1 6A-1 6A1 13A
3 P 1A 2A 1A 1A 4A-1 4A1 2A 2A 4A1 4A-1 4A1 4A-1 13A
13 P 1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5 1A
Type
156.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
156.7.1b R 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1
156.7.1c1 C 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1
156.7.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1
156.7.1d1 C 1 −1 1 1 −i i −1 −1 i −i i −i 1
156.7.1d2 C 1 −1 1 1 i −i −1 −1 −i i −i i 1
156.7.1e1 C 1 1 ζ3−1 ζ3 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1
156.7.1e2 C 1 1 ζ3 ζ3−1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1
156.7.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 −ζ124 ζ122 −ζ125 ζ12 −ζ12 ζ125 1
156.7.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 ζ122 −ζ124 ζ12 −ζ125 ζ125 −ζ12 1
156.7.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 −ζ124 ζ122 ζ125 −ζ12 ζ12 −ζ125 1
156.7.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 ζ122 −ζ124 −ζ12 ζ125 −ζ125 ζ12 1
156.7.12a R 12 0 0 0 0 0 0 0 0 0 0 0 −1

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

Data not computed