Galois group: 9T12

• Label: 9T12
• Degree: $9$
• Order: $54$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $(C_3^2:C_3):C_2$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$ x 4
$18$:	$C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

9T12 x 3, 18T24 x 4, 27T6

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

Group invariants

Order:		$54=2 \cdot 3^{3}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		54.8
Character table:

		1A	2A	3A1	3A-1	3B	3C	3D	3E	6A1	6A-1
	Size	1	9	1	1	6	6	6	6	9	9
	2 P	1A	1A	3A-1	3A1	3B	3C	3D	3E	3A1	3A-1
	3 P	1A	2A	1A	1A	1A	1A	1A	1A	2A	2A
	Type
54.8.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
54.8.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
54.8.2a	R	$2$	$0$	$2$	$2$	$-1$	$-1$	$-1$	$2$	$0$	$0$
54.8.2b	R	$2$	$0$	$2$	$2$	$-1$	$-1$	$2$	$-1$	$0$	$0$
54.8.2c	R	$2$	$0$	$2$	$2$	$-1$	$2$	$-1$	$-1$	$0$	$0$
54.8.2d	R	$2$	$0$	$2$	$2$	$2$	$-1$	$-1$	$-1$	$0$	$0$
54.8.3a1	C	$3$	$1$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	${\zeta }_3$	${\zeta }_3^{-1}$
54.8.3a2	C	$3$	$1$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	${\zeta }_3^{-1}$	${\zeta }_3$
54.8.3b1	C	$3$	$-1$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
54.8.3b2	C	$3$	$-1$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$