Galois group: 13T3

• Label: 13T3
• Degree: $13$
• Order: $39$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{13}:C_3$

Group action invariants

Degree $n$:		$13$
Transitive number $t$:		$3$
Group:		$C_{13}:C_3$
CHM label:		$F_{39}(13)=13:3$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3,4,5,6,7,8,9,10,11,12,13), (1,3,9)(2,6,5)(4,12,10)(7,8,11)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

39T2

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 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 1 $	$13$	$3$	$( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)$
	$ 3, 3, 3, 3, 1 $	$13$	$3$	$( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)$
	$ 13 $	$3$	$13$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)$
	$ 13 $	$3$	$13$	$( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$
	$ 13 $	$3$	$13$	$( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$
	$ 13 $	$3$	$13$	$( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)$

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	$3$	$0$	$0$	${\zeta }_{13}^{-4}+{\zeta }_{13}+{\zeta }_{13}^3$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-1}+{\zeta }_{13}^4$	${\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}$