Galois group: 14T6

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

Group invariants

Abstract group:		$F_8$
Order:		$56=2^{3} \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$14$
Transitive number $t$:		$6$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{14}$	$1$	$1$	$0$	$()$
2A	$2^{4},1^{6}$	$7$	$2$	$4$	$( 3,10)( 5,12)( 6,13)( 7,14)$
7A1	$7^{2}$	$8$	$7$	$12$	$( 1, 6, 4, 9, 7, 5,10)( 2,14,12, 3, 8,13,11)$
7A-1	$7^{2}$	$8$	$7$	$12$	$( 1,10, 5, 7, 9, 4, 6)( 2,11,13, 8, 3,12,14)$
7A2	$7^{2}$	$8$	$7$	$12$	$( 1, 4, 7,10, 6, 9, 5)( 2,12, 8,11,14, 3,13)$
7A-2	$7^{2}$	$8$	$7$	$12$	$( 1, 5, 9, 6,10, 7, 4)( 2,13, 3,14,11, 8,12)$
7A3	$7^{2}$	$8$	$7$	$12$	$( 1, 9,10, 4, 5, 6, 7)( 2, 3,11,12,13,14, 8)$
7A-3	$7^{2}$	$8$	$7$	$12$	$( 1, 7, 6, 5, 4,10, 9)( 2, 8,14,13,12,11, 3)$

Malle's constant $a(G)$:     $1/4$

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

Regular extensions

Data not computed