Galois group: 26T2

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

Group action invariants

Degree $n$:		$26$
Transitive number $t$:		$2$
Group:		$D_{13}$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$26$
Generators:		(1,23)(2,24)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,26), (1,4,6,8,10,12,14,16,18,20,22,24,25)(2,3,5,7,9,11,13,15,17,19,21,23,26)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $D_{13}$

Low degree siblings

13T2

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
	$1^{26}$	$1$	$1$	$()$
	$2^{13}$	$13$	$2$	$( 1, 2)( 3,25)( 4,26)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,19)(11,18)(12,17) (13,16)(14,15)$
	$13^{2}$	$2$	$13$	$( 1, 4, 6, 8,10,12,14,16,18,20,22,24,25)( 2, 3, 5, 7, 9,11,13,15,17,19,21,23, 26)$
	$13^{2}$	$2$	$13$	$( 1, 6,10,14,18,22,25, 4, 8,12,16,20,24)( 2, 5, 9,13,17,21,26, 3, 7,11,15,19, 23)$
	$13^{2}$	$2$	$13$	$( 1, 8,14,20,25, 6,12,18,24, 4,10,16,22)( 2, 7,13,19,26, 5,11,17,23, 3, 9,15, 21)$
	$13^{2}$	$2$	$13$	$( 1,10,18,25, 8,16,24, 6,14,22, 4,12,20)( 2, 9,17,26, 7,15,23, 5,13,21, 3,11, 19)$
	$13^{2}$	$2$	$13$	$( 1,12,22, 6,16,25,10,20, 4,14,24, 8,18)( 2,11,21, 5,15,26, 9,19, 3,13,23, 7, 17)$
	$13^{2}$	$2$	$13$	$( 1,14,25,12,24,10,22, 8,20, 6,18, 4,16)( 2,13,26,11,23, 9,21, 7,19, 5,17, 3, 15)$

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