Properties

Label 13T7
Degree $13$
Order $5616$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSL(3,3)$

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Show commands: Magma

magma: G := TransitiveGroup(13, 7);
 

Group action invariants

Degree $n$:  $13$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(3,3)$
CHM label:   $L(13)=PSL(3,3)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,12)(4,11)(5,6)(7,10)
magma: Generators(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 TypeSizeOrder IndexRepresentative
1A $1^{13}$ $1$ $1$ $0$ $()$
2A $2^{4},1^{5}$ $117$ $2$ $4$ $( 1, 7)( 4, 6)( 8,13)( 9,10)$
3A $3^{3},1^{4}$ $104$ $3$ $6$ $( 1,13, 9)( 2, 5,11)( 7, 8,10)$
3B $3^{4},1$ $624$ $3$ $8$ $( 1,11, 9)( 2, 6, 5)( 3, 7, 8)( 4,13,12)$
4A $4^{2},2^{2},1$ $702$ $4$ $8$ $( 1, 8, 2,11)( 3, 7, 6,13)( 5,12)( 9,10)$
6A $6,3,2,1^{2}$ $936$ $6$ $8$ $( 1,10,13, 7, 9, 8)( 2,11, 5)( 4, 6)$
8A1 $8,4,1$ $702$ $8$ $10$ $( 1,13, 8, 3, 2, 7,11, 6)( 5, 9,12,10)$
8A-1 $8,4,1$ $702$ $8$ $10$ $( 1, 7, 8, 6, 2,13,11, 3)( 5, 9,12,10)$
13A1 $13$ $432$ $13$ $12$ $( 1, 9, 2,12, 3, 7,13, 6, 8, 4,10, 5,11)$
13A-1 $13$ $432$ $13$ $12$ $( 1, 6, 9, 8, 2, 4,12,10, 3, 5, 7,11,13)$
13A2 $13$ $432$ $13$ $12$ $( 1, 7,10, 2, 6,11, 3, 4, 9,13, 5,12, 8)$
13A-2 $13$ $432$ $13$ $12$ $( 1,10, 6, 3, 9, 5, 8, 7, 2,11, 4,13,12)$

magma: ConjugacyClasses(G);
 

Malle's constant $a(G)$:     $1/4$

Group invariants

Order:  $5616=2^{4} \cdot 3^{3} \cdot 13$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  5616.a
magma: IdentifyGroup(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 13A1 13A-2 13A2 13A-1
3 P 1A 2A 1A 1A 4A 2A 8A1 8A-1 13A-2 13A-1 13A1 13A2
13 P 1A 2A 3A 3B 4A 6A 8A-1 8A1 1A 1A 1A 1A
Type

magma: CharacterTable(G);