Galois group: 7T1

• Label: 7T1
• Degree: $7$
• Order: $7$
• Cyclic: yes
• Abelian: yes
• Solvable: yes
• Primitive: yes
• $p$-group: yes
• Group: $C_7$

Group action invariants

Degree $n$:		$7$
Transitive number $t$:		$1$
Group:		$C_7$
CHM label:		$C(7) = 7$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$7$
Generators:		(1,2,3,4,5,6,7)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings 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 $	$1$	$1$	$()$
	$ 7 $	$1$	$7$	$(1,2,3,4,5,6,7)$
	$ 7 $	$1$	$7$	$(1,3,5,7,2,4,6)$
	$ 7 $	$1$	$7$	$(1,4,7,3,6,2,5)$
	$ 7 $	$1$	$7$	$(1,5,2,6,3,7,4)$
	$ 7 $	$1$	$7$	$(1,6,4,2,7,5,3)$
	$ 7 $	$1$	$7$	$(1,7,6,5,4,3,2)$

Group invariants

Order:		$7$ (is prime)
Cyclic:		yes
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$
Label:		7.1
Character table:

		1A	7A1	7A-1	7A2	7A-2	7A3	7A-3
	Size	1	1	1	1	1	1	1
	7 P	1A	7A-3	7A-1	7A3	7A2	7A1	7A-2
	Type
7.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
7.1.1b1	C	$1$	${\zeta }_7^{-3}$	${\zeta }_7^3$	${\zeta }_7$	${\zeta }_7^{-1}$	${\zeta }_7^{-2}$	${\zeta }_7^2$
7.1.1b2	C	$1$	${\zeta }_7^3$	${\zeta }_7^{-3}$	${\zeta }_7^{-1}$	${\zeta }_7$	${\zeta }_7^2$	${\zeta }_7^{-2}$
7.1.1b3	C	$1$	${\zeta }_7^{-2}$	${\zeta }_7^2$	${\zeta }_7^3$	${\zeta }_7^{-3}$	${\zeta }_7$	${\zeta }_7^{-1}$
7.1.1b4	C	$1$	${\zeta }_7^2$	${\zeta }_7^{-2}$	${\zeta }_7^{-3}$	${\zeta }_7^3$	${\zeta }_7^{-1}$	${\zeta }_7$
7.1.1b5	C	$1$	${\zeta }_7^{-1}$	${\zeta }_7$	${\zeta }_7^{-2}$	${\zeta }_7^2$	${\zeta }_7^{-3}$	${\zeta }_7^3$
7.1.1b6	C	$1$	${\zeta }_7$	${\zeta }_7^{-1}$	${\zeta }_7^2$	${\zeta }_7^{-2}$	${\zeta }_7^3$	${\zeta }_7^{-3}$