Galois group: 14T10

• Label: 14T10
• Degree: $14$
• Order: $168$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $\PSL(2,7)$

Group action invariants

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

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $\GL(3,2)$

Low degree siblings

7T5 x 2, 8T37, 14T10, 21T14, 24T284, 28T32, 42T37, 42T38 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$1^{14}$	$1$	$1$	$()$
	$3^{4},1^{2}$	$56$	$3$	$( 2, 3, 6)( 4, 7,12)( 5,11,14)( 9,10,13)$
	$2^{6},1^{2}$	$21$	$2$	$( 2, 4)( 3,10)( 5, 6)( 7,14)( 9,11)(12,13)$
	$7^{2}$	$24$	$7$	$( 1, 2, 3, 7, 6, 4,12)( 5, 8, 9,10,14,13,11)$
	$4^{3},2$	$42$	$4$	$( 1, 2, 8, 9)( 3,12,13, 7)( 4,11)( 5, 6,14,10)$
	$7^{2}$	$24$	$7$	$( 1, 2,12,11,10, 7,13)( 3,14, 6, 8, 9, 5, 4)$

Group invariants

Order:		$168=2^{3} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		168.42
Character table:

		1A	2A	3A	4A	7A1	7A-1
	Size	1	21	56	42	24	24
	2 P	1A	1A	3A	2A	7A1	7A-1
	3 P	1A	2A	1A	4A	7A-1	7A1
	7 P	1A	2A	3A	4A	1A	1A
	Type
168.42.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
168.42.3a1	C	$3$	$-1$	$0$	$1$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
168.42.3a2	C	$3$	$-1$	$0$	$1$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
168.42.6a	R	$6$	$2$	$0$	$0$	$-1$	$-1$
168.42.7a	R	$7$	$-1$	$1$	$-1$	$0$	$0$
168.42.8a	R	$8$	$0$	$-1$	$0$	$1$	$1$