Galois group: 9T1

• Label: 9T1
• Degree: $9$
• Order: $9$
• Cyclic: yes
• Abelian: yes
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_9$

Group action invariants

Degree $n$:		$9$
Transitive number $t$:		$1$
Group:		$C_9$
CHM label:		$C(9)=9$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$9$
Generators:		(1,2,3,4,5,6,7,8,9)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$3$:	$C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

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 $	$1$	$1$	$()$
	$ 9 $	$1$	$9$	$(1,2,3,4,5,6,7,8,9)$
	$ 9 $	$1$	$9$	$(1,3,5,7,9,2,4,6,8)$
	$ 3, 3, 3 $	$1$	$3$	$(1,4,7)(2,5,8)(3,6,9)$
	$ 9 $	$1$	$9$	$(1,5,9,4,8,3,7,2,6)$
	$ 9 $	$1$	$9$	$(1,6,2,7,3,8,4,9,5)$
	$ 3, 3, 3 $	$1$	$3$	$(1,7,4)(2,8,5)(3,9,6)$
	$ 9 $	$1$	$9$	$(1,8,6,4,2,9,7,5,3)$
	$ 9 $	$1$	$9$	$(1,9,8,7,6,5,4,3,2)$

Group invariants

Order:		$9=3^{2}$
Cyclic:		yes
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$
Label:		9.1
Character table:

		1A	3A1	3A-1	9A1	9A-1	9A2	9A-2	9A4	9A-4
	Size	1	1	1	1	1	1	1	1	1
	3 P	1A	3A-1	3A1	9A-1	9A1	9A4	9A2	9A-2	9A-4
	Type
9.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
9.1.1b1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
9.1.1b2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
9.1.1c1	C	$1$	${\zeta }_9^{-3}$	${\zeta }_9^3$	${\zeta }_9^{-4}$	${\zeta }_9^4$	${\zeta }_9$	${\zeta }_9^{-1}$	${\zeta }_9^2$	${\zeta }_9^{-2}$
9.1.1c2	C	$1$	${\zeta }_9^3$	${\zeta }_9^{-3}$	${\zeta }_9^4$	${\zeta }_9^{-4}$	${\zeta }_9^{-1}$	${\zeta }_9$	${\zeta }_9^{-2}$	${\zeta }_9^2$
9.1.1c3	C	$1$	${\zeta }_9^{-3}$	${\zeta }_9^3$	${\zeta }_9^2$	${\zeta }_9^{-2}$	${\zeta }_9^4$	${\zeta }_9^{-4}$	${\zeta }_9^{-1}$	${\zeta }_9$
9.1.1c4	C	$1$	${\zeta }_9^3$	${\zeta }_9^{-3}$	${\zeta }_9^{-2}$	${\zeta }_9^2$	${\zeta }_9^{-4}$	${\zeta }_9^4$	${\zeta }_9$	${\zeta }_9^{-1}$
9.1.1c5	C	$1$	${\zeta }_9^{-3}$	${\zeta }_9^3$	${\zeta }_9^{-1}$	${\zeta }_9$	${\zeta }_9^{-2}$	${\zeta }_9^2$	${\zeta }_9^{-4}$	${\zeta }_9^4$
9.1.1c6	C	$1$	${\zeta }_9^3$	${\zeta }_9^{-3}$	${\zeta }_9$	${\zeta }_9^{-1}$	${\zeta }_9^2$	${\zeta }_9^{-2}$	${\zeta }_9^4$	${\zeta }_9^{-4}$