Galois group: 14T11

• Label: 14T11
• Degree: $14$
• Order: $168$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_8:C_3$

Group invariants

Abstract group:		$F_8:C_3$
Order:		$168=2^{3} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$14$
Transitive number $t$:		$11$
CHM label:		$[2^{3}]F_{21}(7)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,9,11)(2,4,8)(3,13,5)(6,12,10)$, $(3,10)(5,12)(6,13)(7,14)$, $(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$3$:	$C_3$
$21$:	$C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7:C_3$

Low degree siblings

8T36, 24T283, 28T27, 42T26

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	$2^{4},1^{6}$	$7$	$2$	$4$	$( 1, 8)( 3,10)( 4,11)( 5,12)$
3A1	$3^{4},1^{2}$	$28$	$3$	$8$	$( 1,11, 3)( 2, 6,14)( 4,10, 8)( 7, 9,13)$
3A-1	$3^{4},1^{2}$	$28$	$3$	$8$	$( 1, 3,11)( 2,14, 6)( 4, 8,10)( 7,13, 9)$
6A1	$6,3^{2},2$	$28$	$6$	$10$	$( 1,10,11, 8, 3, 4)( 2,14, 6)( 5,12)( 7,13, 9)$
6A-1	$6,3^{2},2$	$28$	$6$	$10$	$( 1, 4, 3, 8,11,10)( 2, 6,14)( 5,12)( 7, 9,13)$
7A1	$7^{2}$	$24$	$7$	$12$	$( 1,10, 5,14, 2, 4,13)( 3,12, 7, 9,11, 6, 8)$
7A-1	$7^{2}$	$24$	$7$	$12$	$( 1,13, 4, 2,14, 5,10)( 3, 8, 6,11, 9, 7,12)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	3A1	3A-1	6A1	6A-1	7A1	7A-1
	Size	1	7	28	28	28	28	24	24
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	7A1	7A-1
	3 P	1A	2A	1A	1A	2A	2A	7A-1	7A1
	7 P	1A	2A	3A1	3A-1	6A1	6A-1	1A	1A
	Type
168.43.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
168.43.1b1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
168.43.1b2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
168.43.3a1	C	$3$	$3$	$0$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
168.43.3a2	C	$3$	$3$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
168.43.7a	R	$7$	$-1$	$1$	$1$	$-1$	$-1$	$0$	$0$
168.43.7b1	C	$7$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	$0$
168.43.7b2	C	$7$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$0$

Regular extensions

Data not computed