Galois group: 14T2

• Label: 14T2
• Degree: $14$
• Order: $14$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Transitivity: $1$
• Primitive: no
• $p$-group: no
• Group: $D_{7}$

Group invariants

Abstract group:		$D_{7}$
Order:		$14=2 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$14$
Transitive number $t$:		$2$
CHM label:		$D_{14}(14)=[7]2$
Parity:		$-1$
Transitivity:		1
Primitive:		no
$\card{\Aut(F/K)}$:		$14$
Generators:		$(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)$, $(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $D_{7}$

Low degree siblings

7T2

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	$1^{14}$	$1$	$1$	$0$	$()$
2A	$2^{7}$	$7$	$2$	$7$	$( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)$
7A1	$7^{2}$	$2$	$7$	$12$	$( 1, 9, 3,11, 5,13, 7)( 2,10, 4,12, 6,14, 8)$
7A2	$7^{2}$	$2$	$7$	$12$	$( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
7A3	$7^{2}$	$2$	$7$	$12$	$( 1,11, 7, 3,13, 9, 5)( 2,12, 8, 4,14,10, 6)$

Malle's constant $a(G)$:     $1/7$

Character table

		1A	2A	7A1	7A2	7A3
	Size	1	7	2	2	2
	2 P	1A	1A	7A2	7A3	7A1
	7 P	1A	2A	7A3	7A1	7A2
	Type
14.1.1a	R	$1$	$1$	$1$	$1$	$1$
14.1.1b	R	$1$	$-1$	$1$	$1$	$1$
14.1.2a1	R	$2$	$0$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$
14.1.2a2	R	$2$	$0$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$
14.1.2a3	R	$2$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$

Regular extensions

Data not computed