Galois group: 13T2

• Label: 13T2
• Degree: $13$
• Order: $26$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $D_{13}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

26T2

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

Group invariants

Order:		$26=2 \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		26.1
Character table:

		1A	2A	13A1	13A2	13A3	13A4	13A5	13A6
	Size	1	13	2	2	2	2	2	2
	2 P	1A	1A	13A1	13A4	13A3	13A6	13A2	13A5
	13 P	1A	2A	13A5	13A6	13A2	13A4	13A3	13A1
	Type
26.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
26.1.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
26.1.2a1	R	$2$	$0$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$	${\zeta }_{13}^{-1}+{\zeta }_{13}$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$
26.1.2a2	R	$2$	$0$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$	${\zeta }_{13}^{-1}+{\zeta }_{13}$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$
26.1.2a3	R	$2$	$0$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$	${\zeta }_{13}^{-1}+{\zeta }_{13}$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$
26.1.2a4	R	$2$	$0$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$	${\zeta }_{13}^{-1}+{\zeta }_{13}$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$
26.1.2a5	R	$2$	$0$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$	${\zeta }_{13}^{-1}+{\zeta }_{13}$
26.1.2a6	R	$2$	$0$	${\zeta }_{13}^{-1}+{\zeta }_{13}$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$