Galois group: 26T6

• Label: 26T6
• Degree: $26$
• Order: $78$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{13}:C_6$

Group invariants

Abstract group:		$C_{13}:C_6$
Order:		$78=2 \cdot 3 \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $C_{13}:C_6$

Low degree siblings

13T5, 39T6

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	$2^{13}$	$13$	$2$	$13$	$( 1,19)( 2,20)( 3,17)( 4,18)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)(21,26)(22,25)(23,24)$
3A1	$3^{8},1^{2}$	$13$	$3$	$16$	$( 1, 8, 9)( 2, 7,10)( 3,26,16)( 4,25,15)( 5,18,22)( 6,17,21)(11,20,13)(12,19,14)$
3A-1	$3^{8},1^{2}$	$13$	$3$	$16$	$( 1, 9, 8)( 2,10, 7)( 3,16,26)( 4,15,25)( 5,22,18)( 6,21,17)(11,13,20)(12,14,19)$
6A1	$6^{4},2$	$13$	$6$	$21$	$( 1,12, 8,19, 9,14)( 2,11, 7,20,10,13)( 3, 6,26,17,16,21)( 4, 5,25,18,15,22)(23,24)$
6A-1	$6^{4},2$	$13$	$6$	$21$	$( 1,14, 9,19, 8,12)( 2,13,10,20, 7,11)( 3,21,16,17,26, 6)( 4,22,15,18,25, 5)(23,24)$
13A1	$13^{2}$	$6$	$13$	$24$	$( 1, 9,17,25, 8,15,23, 6,13,21, 4,11,20)( 2,10,18,26, 7,16,24, 5,14,22, 3,12,19)$
13A2	$13^{2}$	$6$	$13$	$24$	$( 1,17, 8,23,13, 4,20, 9,25,15, 6,21,11)( 2,18, 7,24,14, 3,19,10,26,16, 5,22,12)$

Malle's constant $a(G)$:     $1/13$

Character table

		1A	2A	3A1	3A-1	6A1	6A-1	13A1	13A2
	Size	1	13	13	13	13	13	6	6
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	13A2	13A1
	3 P	1A	2A	1A	1A	2A	2A	13A1	13A2
	13 P	1A	2A	3A1	3A-1	6A1	6A-1	1A	1A
	Type
78.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
78.1.1b	R	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$
78.1.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
78.1.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
78.1.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$
78.1.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$
78.1.6a1	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$	$-{\zeta }_{13}^{-6}-{\zeta }_{13}^{-5}-{\zeta }_{13}^{-2}-1-{\zeta }_{13}^2-{\zeta }_{13}^5-{\zeta }_{13}^6$
78.1.6a2	R	$6$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_{13}^{-6}-{\zeta }_{13}^{-5}-{\zeta }_{13}^{-2}-1-{\zeta }_{13}^2-{\zeta }_{13}^5-{\zeta }_{13}^6$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$