Galois group: 39T43

• Label: 39T43
• Degree: $39$
• Order: $5616$
• Cyclic: no
• Abelian: no
• Solvable: no
• Transitivity: $1$
• Primitive: no
• $p$-group: no
• Group: $\PSL(3,3)$

Group invariants

Abstract group:		$\PSL(3,3)$
Order:		$5616=2^{4} \cdot 3^{3} \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

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

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

Malle's constant $a(G)$:     $1/16$

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	$16$	$0$	$-2$	$1$	$0$	$0$	$0$	$0$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}$	${\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$	${\zeta }_{13}^{-4}+{\zeta }_{13}+{\zeta }_{13}^3$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-1}+{\zeta }_{13}^4$
5616.a.16a2	C	$16$	$0$	$-2$	$1$	$0$	$0$	$0$	$0$	${\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-1}+{\zeta }_{13}^4$	${\zeta }_{13}^{-4}+{\zeta }_{13}+{\zeta }_{13}^3$
5616.a.16a3	C	$16$	$0$	$-2$	$1$	$0$	$0$	$0$	$0$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-1}+{\zeta }_{13}^4$	${\zeta }_{13}^{-4}+{\zeta }_{13}+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}$	${\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$
5616.a.16a4	C	$16$	$0$	$-2$	$1$	$0$	$0$	$0$	$0$	${\zeta }_{13}^{-4}+{\zeta }_{13}+{\zeta }_{13}^3$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-1}+{\zeta }_{13}^4$	${\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}$
5616.a.26a	R	$26$	$2$	$-1$	$-1$	$2$	$-1$	$0$	$0$	$0$	$0$	$0$	$0$
5616.a.26b1	C	$26$	$-2$	$-1$	$-1$	$0$	$1$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	$0$	$0$	$0$	$0$
5616.a.26b2	C	$26$	$-2$	$-1$	$-1$	$0$	$1$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	$0$	$0$	$0$	$0$
5616.a.27a	R	$27$	$3$	$0$	$0$	$-1$	$0$	$-1$	$-1$	$1$	$1$	$1$	$1$
5616.a.39a	R	$39$	$-1$	$3$	$0$	$-1$	$-1$	$1$	$1$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed