Galois group: 7T2

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

Group action invariants

Degree $n$:		$7$
Transitive number $t$:		$2$
Group:		$D_{7}$
CHM label:		$D(7) = 7:2$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,6)(2,5)(3,4), (1,2,3,4,5,6,7)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

14T2

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 $	$1$	$1$	$()$
	$ 2, 2, 2, 1 $	$7$	$2$	$(2,7)(3,6)(4,5)$
	$ 7 $	$2$	$7$	$(1,2,3,4,5,6,7)$
	$ 7 $	$2$	$7$	$(1,3,5,7,2,4,6)$
	$ 7 $	$2$	$7$	$(1,4,7,3,6,2,5)$

Group invariants

Order:		$14=2 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		14.1
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	$2$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$