Galois group: 14T3

• Label: 14T3
• Degree: $14$
• Order: $28$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{14}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$14$:	$D_{7}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $D_{7}$

Low degree siblings

14T3, 28T4

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 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 1, 1 $	$7$	$2$	$( 2,14)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)$
	$ 2, 2, 2, 2, 2, 2, 2 $	$7$	$2$	$( 1, 2)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$
	$ 14 $	$2$	$14$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
	$ 7, 7 $	$2$	$7$	$( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
	$ 14 $	$2$	$14$	$( 1, 4, 7,10,13, 2, 5, 8,11,14, 3, 6, 9,12)$
	$ 7, 7 $	$2$	$7$	$( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$
	$ 14 $	$2$	$14$	$( 1, 6,11, 2, 7,12, 3, 8,13, 4, 9,14, 5,10)$
	$ 7, 7 $	$2$	$7$	$( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$
	$ 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$

Group invariants

Order:		$28=2^{2} \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		28.3
Character table:

		1A	2A	2B	2C	7A1	7A2	7A3	14A1	14A3	14A5
	Size	1	1	7	7	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	7A1	7A2	7A3	7A3	7A2	7A1
	7 P	1A	2A	2B	2C	7A2	7A3	7A1	14A5	14A1	14A3
	Type
28.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
28.3.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
28.3.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
28.3.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
28.3.2a1	R	$2$	$2$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$
28.3.2a2	R	$2$	$2$	$0$	$0$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$
28.3.2a3	R	$2$	$2$	$0$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$
28.3.2b1	R	$2$	$-2$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$
28.3.2b2	R	$2$	$-2$	$0$	$0$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$
28.3.2b3	R	$2$	$-2$	$0$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$