Galois group: 14T4

• Label: 14T4
• Degree: $14$
• Order: $42$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_7$

Group action invariants

Degree $n$:		$14$
Transitive number $t$:		$4$
Group:		$F_7$
CHM label:		$2[1/2]F_{42}(7)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,6)(2,5)(3,4)(7,14)(8,13)(9,12)(10,11), (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)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $F_7$

Low degree siblings

7T4, 21T4, 42T4

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

Group invariants

Order:		$42=2 \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		42.1
Character table:

		1A	2A	3A1	3A-1	6A1	6A-1	7A
	Size	1	7	7	7	7	7	6
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	7A
	3 P	1A	2A	1A	1A	2A	2A	7A
	7 P	1A	2A	3A1	3A-1	6A1	6A-1	1A
	Type
42.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
42.1.1b	R	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
42.1.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$
42.1.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$
42.1.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$
42.1.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$
42.1.6a	R	$6$	$0$	$0$	$0$	$0$	$0$	$-1$