Galois group: 7T2

• Label: 7T2
• Degree: $7$
• Order: $14$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-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 $n$:		$7$
Transitive number $t$:		$2$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{7}$	$1$	$1$	$0$	$()$
2A	$2^{3},1$	$7$	$2$	$3$	$(1,6)(2,5)(3,4)$
7A1	$7$	$2$	$7$	$6$	$(1,5,2,6,3,7,4)$
7A2	$7$	$2$	$7$	$6$	$(1,2,3,4,5,6,7)$
7A3	$7$	$2$	$7$	$6$	$(1,6,4,2,7,5,3)$

Malle's constant $a(G)$:     $1/3$

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

$f_{ 1 } =$

$x^{7} - x^{6} - t x^{5} + 6 x^{4} + \left(2 t + 39\right) x^{3} + 17 x^{2} - t x + 2$