Galois group: 26T2

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

Group invariants

Abstract group:		$D_{13}$
Order:		$26=2 \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$26$
Transitive number $t$:		$2$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{26}$	$1$	$1$	$0$	$()$
2A	$2^{13}$	$13$	$2$	$13$	$( 1,26)( 2,25)( 3,24)( 4,23)( 5,22)( 6,21)( 7,20)( 8,19)( 9,18)(10,17)(11,16)(12,15)(13,14)$
13A1	$13^{2}$	$2$	$13$	$24$	$( 1,20,12, 4,22,14, 6,24,16, 8,25,18,10)( 2,19,11, 3,21,13, 5,23,15, 7,26,17, 9)$
13A2	$13^{2}$	$2$	$13$	$24$	$( 1,12,22, 6,16,25,10,20, 4,14,24, 8,18)( 2,11,21, 5,15,26, 9,19, 3,13,23, 7,17)$
13A3	$13^{2}$	$2$	$13$	$24$	$( 1, 4, 6, 8,10,12,14,16,18,20,22,24,25)( 2, 3, 5, 7, 9,11,13,15,17,19,21,23,26)$
13A4	$13^{2}$	$2$	$13$	$24$	$( 1,22,16,10, 4,24,18,12, 6,25,20,14, 8)( 2,21,15, 9, 3,23,17,11, 5,26,19,13, 7)$
13A5	$13^{2}$	$2$	$13$	$24$	$( 1,14,25,12,24,10,22, 8,20, 6,18, 4,16)( 2,13,26,11,23, 9,21, 7,19, 5,17, 3,15)$
13A6	$13^{2}$	$2$	$13$	$24$	$( 1, 6,10,14,18,22,25, 4, 8,12,16,20,24)( 2, 5, 9,13,17,21,26, 3, 7,11,15,19,23)$

Malle's constant $a(G)$:     $1/13$

Character table

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

Regular extensions

Data not computed