Galois group: 13T4

• Label: 13T4
• Degree: $13$
• Order: $52$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{13}:C_4$

Group action invariants

Degree $n$:		$13$
Transitive number $t$:		$4$
Group:		$C_{13}:C_4$
CHM label:		$F_{52}(13)=13:4$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3,4,5,6,7,8,9,10,11,12,13), (1,5,12,8)(2,10,11,3)(4,7,9,6)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

26T4

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

Group invariants

Order:		$52=2^{2} \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		52.3
Character table:

		1A	2A	4A1	4A-1	13A1	13A2	13A4
	Size	1	13	13	13	4	4	4
	2 P	1A	1A	2A	2A	13A2	13A4	13A1
	13 P	1A	2A	4A-1	4A1	13A2	13A4	13A1
	Type
52.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
52.3.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
52.3.1c1	C	$1$	$-1$	$-i$	$i$	$1$	$1$	$1$
52.3.1c2	C	$1$	$-1$	$i$	$-i$	$1$	$1$	$1$
52.3.4a1	R	$4$	$0$	$0$	$0$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-4}+{\zeta }_{13}^4+{\zeta }_{13}^6$	${\zeta }_{13}^{-5}+{\zeta }_{13}^{-1}+{\zeta }_{13}+{\zeta }_{13}^5$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^3$
52.3.4a2	R	$4$	$0$	$0$	$0$	${\zeta }_{13}^{-5}+{\zeta }_{13}^{-1}+{\zeta }_{13}+{\zeta }_{13}^5$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-4}+{\zeta }_{13}^4+{\zeta }_{13}^6$
52.3.4a3	R	$4$	$0$	$0$	$0$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-4}+{\zeta }_{13}^4+{\zeta }_{13}^6$	${\zeta }_{13}^{-5}+{\zeta }_{13}^{-1}+{\zeta }_{13}+{\zeta }_{13}^5$