LMFDB - Galois group: 39T43

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

Malle's constant $a(G)$: $1/16$

magma: ConjugacyClasses(G);

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: