Galois group: 7T4

• Label: 7T4
• Degree: $7$
• Order: $42$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $F_7$

Group action invariants

Degree $n$:		$7$
Transitive number $t$:		$4$
Group:		$F_7$
CHM label:		$F_{42}(7) = 7:6$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,3,2,6,4,5), (1,2,3,4,5,6,7)

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

Prime degree - none

Low degree siblings

14T4, 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 $	$1$	$1$	$()$
	$ 3, 3, 1 $	$7$	$3$	$(2,3,5)(4,7,6)$
	$ 6, 1 $	$7$	$6$	$(2,4,3,7,5,6)$
	$ 3, 3, 1 $	$7$	$3$	$(2,5,3)(4,6,7)$
	$ 6, 1 $	$7$	$6$	$(2,6,5,7,3,4)$
	$ 2, 2, 2, 1 $	$7$	$2$	$(2,7)(3,6)(4,5)$
	$ 7 $	$6$	$7$	$(1,2,3,4,5,6,7)$

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	$6$	$0$	$0$	$0$	$0$	$0$	$-1$