Galois group: 13T5

• Label: 13T5
• Degree: $13$
• Order: $78$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{13}:C_6$

Group action invariants

Degree $n$:		$13$
Transitive number $t$:		$5$
Group:		$C_{13}:C_6$
CHM label:		$F_{78}(13)=13:6$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3,4,5,6,7,8,9,10,11,12,13), (1,4,3,12,9,10)(2,8,6,11,5,7)

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

Prime degree - none

Low degree siblings

26T6, 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 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 1 $	$13$	$3$	$( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)$
	$ 6, 6, 1 $	$13$	$6$	$( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)$
	$ 3, 3, 3, 3, 1 $	$13$	$3$	$( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)$
	$ 6, 6, 1 $	$13$	$6$	$( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)$
	$ 2, 2, 2, 2, 2, 2, 1 $	$13$	$2$	$( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
	$ 13 $	$6$	$13$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)$
	$ 13 $	$6$	$13$	$( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$

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