# Properties

 Label 39T43 Degree $39$ Order $5616$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $\PSL(3,3)$

Show commands: Magma

magma: G := TransitiveGroup(39, 43);

## Group action invariants

 Degree $n$: $39$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $43$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $\PSL(3,3)$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,8,18,13,5,20,37,36)(2,9,16,14,4,19,39,34)(3,7,17,15,6,21,38,35)(10,24,33,27)(11,23,32,26,12,22,31,25)(28,30), (1,37,25,7)(2,39,26,9,3,38,27,8)(4,13,22,35,30,19,33,18)(5,14,23,36,28,20,31,16)(6,15,24,34,29,21,32,17)(11,12) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: None

Degree 13: $\PSL(3,3)$

## Low degree siblings

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

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 $1^{39}$ $1$ $1$ $0$ $()$ 2A $2^{16},1^{7}$ $117$ $2$ $16$ $( 1, 2)( 4, 5)( 7,19)( 8,20)( 9,21)(11,12)(13,17)(14,18)(15,16)(25,31)(26,32)(27,33)(28,29)(34,37)(35,39)(36,38)$ 3A $3^{12},1^{3}$ $104$ $3$ $24$ $( 4,11,28)( 5,12,29)( 6,10,30)( 7,38,27)( 8,37,26)( 9,39,25)(13,14,15)(16,17,18)(19,36,33)(20,34,32)(21,35,31)(22,24,23)$ 3B $3^{13}$ $624$ $3$ $26$ $( 1,36,31)( 2,34,32)( 3,35,33)( 4, 7,37)( 5, 8,39)( 6, 9,38)(10,17,30)(11,18,28)(12,16,29)(13,25,24)(14,27,23)(15,26,22)(19,20,21)$ 4A $4^{8},2^{2},1^{3}$ $702$ $4$ $26$ $( 1,35,16,26)( 2,36,17,25)( 3,34,18,27)( 4,38,31,12)( 5,39,32,11)( 6,37,33,10)( 7,20)( 8,19, 9,21)(22,29)(23,28,24,30)$ 6A $6^{5},3^{2},2,1$ $936$ $6$ $30$ $( 1, 2)( 4,29,11, 5,28,12)( 6,30,10)( 7,33,38,19,27,36)( 8,32,37,20,26,34)( 9,31,39,21,25,35)(13,16,14,17,15,18)(22,23,24)$ 8A1 $8^{4},4,2,1$ $702$ $8$ $32$ $( 1,11,35, 5,16,39,26,32)( 2,12,36, 4,17,38,25,31)( 3,10,34, 6,18,37,27,33)( 7,22,20,29)( 8,23,19,28, 9,24,21,30)(13,15)$ 8A-1 $8^{4},4,2,1$ $702$ $8$ $32$ $( 1,39,35,32,16,11,26, 5)( 2,38,36,31,17,12,25, 4)( 3,37,34,33,18,10,27, 6)( 7,22,20,29)( 8,24,19,30, 9,23,21,28)(13,15)$ 13A1 $13^{3}$ $432$ $13$ $36$ $( 1,36,25,17,31,14,20,30, 6,39, 7,11,23)( 2,34,27,16,32,13,21,28, 4,38, 9,12,22)( 3,35,26,18,33,15,19,29, 5,37, 8,10,24)$ 13A-1 $13^{3}$ $432$ $13$ $36$ $( 1,30,36, 6,25,39,17, 7,31,11,14,23,20)( 2,28,34, 4,27,38,16, 9,32,12,13,22,21)( 3,29,35, 5,26,37,18, 8,33,10,15,24,19)$ 13A2 $13^{3}$ $432$ $13$ $36$ $( 1,14, 7,25,30,23,31,39,36,20,11,17, 6)( 2,13, 9,27,28,22,32,38,34,21,12,16, 4)( 3,15, 8,26,29,24,33,37,35,19,10,18, 5)$ 13A-2 $13^{3}$ $432$ $13$ $36$ $( 1, 7,30,31,36,11, 6,14,25,23,39,20,17)( 2, 9,28,32,34,12, 4,13,27,22,38,21,16)( 3, 8,29,33,35,10, 5,15,26,24,37,19,18)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/16$

## 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);