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 invariants

Abstract group:		$C_{13}:C_4$
Order:		$52=2^{2} \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$13$
Transitive number $t$:		$4$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{13}$	$1$	$1$	$0$	$()$
2A	$2^{6},1$	$13$	$2$	$6$	$( 1, 6)( 2, 5)( 3, 4)( 7,13)( 8,12)( 9,11)$
4A1	$4^{3},1$	$13$	$4$	$9$	$( 1, 3, 6, 4)( 2,11, 5, 9)( 7,12,13, 8)$
4A-1	$4^{3},1$	$13$	$4$	$9$	$( 1, 4, 6, 3)( 2, 9, 5,11)( 7, 8,13,12)$
13A1	$13$	$4$	$13$	$12$	$( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
13A2	$13$	$4$	$13$	$12$	$( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)$
13A4	$13$	$4$	$13$	$12$	$( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)$

Malle's constant $a(G)$:     $1/6$

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	$4$	$0$	$0$	$0$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-4}+{\zeta }_{13}^4+{\zeta }_{13}^6$	${\zeta }_{13}^{-5}+{\zeta }_{13}^{-1}+{\zeta }_{13}+{\zeta }_{13}^5$

Regular extensions

Data not computed