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 action invariants

Degree $n$:		$26$
Transitive number $t$:		$6$
Group:		$C_{13}:C_6$
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

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

Group invariants

Order:		$78=2 \cdot 3 \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		78.1
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	$6$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_{13}^{-6}-{\zeta }_{13}^{-5}-{\zeta }_{13}^{-2}-1-{\zeta }_{13}^2-{\zeta }_{13}^5-{\zeta }_{13}^6$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$