Galois group: 9T3

• Label: 9T3
• Degree: $9$
• Order: $18$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{9}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

18T5

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

Group invariants

Order:		$18=2 \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		18.1
Character table:

		1A	2A	3A	9A1	9A2	9A4
	Size	1	9	2	2	2	2
	2 P	1A	1A	3A	9A2	9A4	9A1
	3 P	1A	2A	1A	3A	3A	3A
	Type
18.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
18.1.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$
18.1.2a	R	$2$	$0$	$2$	$-1$	$-1$	$-1$
18.1.2b1	R	$2$	$0$	$-1$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-2}+{\zeta }_9^2$
18.1.2b2	R	$2$	$0$	$-1$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-1}+{\zeta }_9$
18.1.2b3	R	$2$	$0$	$-1$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-4}+{\zeta }_9^4$