Galois group: 14T6

• Label: 14T6
• Degree: $14$
• Order: $56$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_8$

Group action invariants

Degree $n$:		$14$
Transitive number $t$:		$6$
Group:		$F_8$
CHM label:		$[2^{3}]7$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(3,10)(5,12)(6,13)(7,14), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$7$:	$C_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7$

Low degree siblings

8T25, 28T11

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

Group invariants

Order:		$56=2^{3} \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		56.11
Character table:

		1A	2A	7A1	7A-1	7A2	7A-2	7A3	7A-3
	Size	1	7	8	8	8	8	8	8
	2 P	1A	1A	7A2	7A-1	7A-2	7A-3	7A1	7A3
	7 P	1A	2A	7A3	7A2	7A-3	7A-1	7A-2	7A1
	Type
56.11.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
56.11.1b1	C	$1$	$1$	${\zeta }_7^{-3}$	${\zeta }_7^3$	${\zeta }_7$	${\zeta }_7^{-1}$	${\zeta }_7^{-2}$	${\zeta }_7^2$
56.11.1b2	C	$1$	$1$	${\zeta }_7^3$	${\zeta }_7^{-3}$	${\zeta }_7^{-1}$	${\zeta }_7$	${\zeta }_7^2$	${\zeta }_7^{-2}$
56.11.1b3	C	$1$	$1$	${\zeta }_7^{-2}$	${\zeta }_7^2$	${\zeta }_7^3$	${\zeta }_7^{-3}$	${\zeta }_7$	${\zeta }_7^{-1}$
56.11.1b4	C	$1$	$1$	${\zeta }_7^2$	${\zeta }_7^{-2}$	${\zeta }_7^{-3}$	${\zeta }_7^3$	${\zeta }_7^{-1}$	${\zeta }_7$
56.11.1b5	C	$1$	$1$	${\zeta }_7^{-1}$	${\zeta }_7$	${\zeta }_7^{-2}$	${\zeta }_7^2$	${\zeta }_7^{-3}$	${\zeta }_7^3$
56.11.1b6	C	$1$	$1$	${\zeta }_7$	${\zeta }_7^{-1}$	${\zeta }_7^2$	${\zeta }_7^{-2}$	${\zeta }_7^3$	${\zeta }_7^{-3}$
56.11.7a	R	$7$	$-1$	$0$	$0$	$0$	$0$	$0$	$0$