Properties

Label 13T7
13T7 1 2 1->2 3 2->3 12 2->12 4 3->4 5 4->5 11 4->11 6 5->6 5->6 7 6->7 8 7->8 10 7->10 9 8->9 9->10 10->11 11->12 13 12->13 13->1
Degree $13$
Order $5616$
Cyclic no
Abelian no
Solvable no
Transitivity $2$
Primitive yes
$p$-group no
Group: $\PSL(3,3)$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(13, 7);
 
Copy content sage:G = TransitiveGroup(13, 7)
 
Copy content oscar:G = transitive_group(13, 7)
 
Copy content gap:G := TransitiveGroup(13, 7);
 

Group invariants

Abstract group:  $\PSL(3,3)$
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:  $5616=2^{4} \cdot 3^{3} \cdot 13$
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:  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 $n$:  $13$
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 $t$:  $7$
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:   $L(13)=PSL(3,3)$
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:  $(1,2,3,4,5,6,7,8,9,10,11,12,13)$, $(2,12)(4,11)(5,6)(7,10)$
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

Prime degree - none

Low degree siblings

13T7, 26T39 x 2, 39T43 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{13}$ $1$ $1$ $0$ $()$
2A $2^{4},1^{5}$ $117$ $2$ $4$ $( 3,13)( 5,12)( 6, 7)( 8,11)$
3A $3^{3},1^{4}$ $104$ $3$ $6$ $( 1, 6, 8)( 4, 7,13)( 9,12,10)$
3B $3^{4},1$ $624$ $3$ $8$ $( 1,12,13)( 2, 3,11)( 4, 6,10)( 7, 8, 9)$
4A $4^{2},2^{2},1$ $702$ $4$ $8$ $( 1,10)( 2, 4)( 3,12,13, 5)( 6, 8, 7,11)$
6A $6,3,2,1^{2}$ $936$ $6$ $8$ $( 1, 8, 6)( 2, 3)( 4,10, 7, 9,13,12)$
8A1 $8,4,1$ $702$ $8$ $10$ $( 1, 4,10, 2)( 3,11,12, 6,13, 8, 5, 7)$
8A-1 $8,4,1$ $702$ $8$ $10$ $( 1, 2,10, 4)( 3, 7, 5, 8,13, 6,12,11)$
13A1 $13$ $432$ $13$ $12$ $( 1, 2, 6,10, 9,13,12, 3, 8, 4, 7,11, 5)$
13A-1 $13$ $432$ $13$ $12$ $( 1, 5,11, 7, 4, 8, 3,12,13, 9,10, 6, 2)$
13A2 $13$ $432$ $13$ $12$ $( 1, 6, 9,12, 8, 7, 5, 2,10,13, 3, 4,11)$
13A-2 $13$ $432$ $13$ $12$ $( 1,11, 4, 3,13,10, 2, 5, 7, 8,12, 9, 6)$

Malle's constant $a(G)$:     $1/4$

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 6A 8A1 8A-1 13A1 13A-1 13A2 13A-2
Size 1 117 104 624 702 936 702 702 432 432 432 432
2 P 1A 1A 3A 3B 2A 3A 4A 4A 13A2 13A-2 13A-1 13A1
3 P 1A 2A 1A 1A 4A 2A 8A1 8A-1 13A1 13A-1 13A2 13A-2
13 P 1A 2A 3A 3B 4A 6A 8A-1 8A1 1A 1A 1A 1A
Type
5616.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1
5616.a.12a R 12 4 3 0 0 1 0 0 −1 −1 −1 −1
5616.a.13a R 13 −3 4 1 1 0 −1 −1 0 0 0 0
5616.a.16a1 C 16 0 −2 1 0 0 0 0 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134
5616.a.16a2 C 16 0 −2 1 0 0 0 0 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133
5616.a.16a3 C 16 0 −2 1 0 0 0 0 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136
5616.a.16a4 C 16 0 −2 1 0 0 0 0 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2
5616.a.26a R 26 2 −1 −1 2 −1 0 0 0 0 0 0
5616.a.26b1 C 26 −2 −1 −1 0 1 −ζ8−ζ83 ζ8+ζ83 0 0 0 0
5616.a.26b2 C 26 −2 −1 −1 0 1 ζ8+ζ83 −ζ8−ζ83 0 0 0 0
5616.a.27a R 27 3 0 0 −1 0 −1 −1 1 1 1 1
5616.a.39a R 39 −1 3 0 −1 −1 1 1 0 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

$f_{ 1 } =$ $\left(x^{3} - 18 x^{2} + 54 x - 108\right) \left(x^{4} + 8 x^{3} - 108 x^{2} + 432 x - 540\right) \left(x^{6} - 6 x^{4} + 64 x^{3} - 36 x^{2} + 216\right)-\left(3 x^{4} - 28 x^{3} + 108 x^{2} - 216 x + 108\right)^{2} \left(x^{4} + 8 x^{3} + 108\right) t$ Copy content Toggle raw display