Galois group: 39T2

• Label: 39T2
• Degree: $39$
• Order: $39$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{13}:C_3$

Group action invariants

Degree $n$:		$39$
Transitive number $t$:		$2$
Group:		$C_{13}:C_3$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$39$
Generators:		(1,14,6)(2,15,4)(3,13,5)(7,30,23)(8,28,24)(9,29,22)(10,16,31)(11,17,32)(12,18,33)(19,20,21)(25,35,37)(26,36,38)(27,34,39), (1,3,2)(4,11,30)(5,12,28)(6,10,29)(7,21,18)(8,19,16)(9,20,17)(13,37,32)(14,38,33)(15,39,31)(22,25,36)(23,26,34)(24,27,35)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$3$:	$C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 13: $C_{13}:C_3$

Low degree siblings

13T3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

Group invariants

Order:		$39=3 \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		39.1
Character table:

		1A	3A1	3A-1	13A1	13A-1	13A2	13A-2
	Size	1	13	13	3	3	3	3
	3 P	1A	3A-1	3A1	13A2	13A-2	13A-1	13A1
	13 P	1A	1A	1A	13A1	13A-1	13A2	13A-2
	Type
39.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
39.1.1b1	C	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$
39.1.1b2	C	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$
39.1.3a1	C	$3$	$0$	$0$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}$	${\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$	${\zeta }_{13}^{-4}+{\zeta }_{13}+{\zeta }_{13}^3$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-1}+{\zeta }_{13}^4$
39.1.3a2	C	$3$	$0$	$0$	${\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-1}+{\zeta }_{13}^4$	${\zeta }_{13}^{-4}+{\zeta }_{13}+{\zeta }_{13}^3$
39.1.3a3	C	$3$	$0$	$0$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-1}+{\zeta }_{13}^4$	${\zeta }_{13}^{-4}+{\zeta }_{13}+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}$	${\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$
39.1.3a4	C	$3$	$0$	$0$	${\zeta }_{13}^{-4}+{\zeta }_{13}+{\zeta }_{13}^3$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-1}+{\zeta }_{13}^4$	${\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}$