Properties

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

Related objects

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

 Label Cycle Type Size Order Index Representative 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);