Galois group: 9T17

• Label: 9T17
• Degree: $9$
• Order: $81$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_3 \wr C_3 $

Group action invariants

Degree $n$:		$9$
Transitive number $t$:		$17$
Group:		$C_3 \wr C_3 $
CHM label:		$[3^{3}]3=3wr3$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		(1,2,9), (1,4,7)(2,5,8)(3,6,9)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$3$:	$C_3$ x 4
$9$:	$C_3^2$
$27$:	$C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

9T17 x 2, 27T19, 27T21, 27T27 x 3

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

Group invariants

Order:		$81=3^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$
Label:		81.7
Character table:

		1A	3A1	3A-1	3B1	3B-1	3C1	3C-1	3D1	3D-1	3E1	3E-1	3F1	3F-1	9A1	9A-1	9B1	9B-1
	Size	1	1	1	3	3	3	3	3	3	3	3	9	9	9	9	9	9
	3 P	1A	3A-1	3A1	3B1	3E1	3C-1	3C1	3D1	3E-1	3D-1	3B-1	3F-1	3F1	9A-1	9A1	9B-1	9B1
	Type
81.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
81.7.1b1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
81.7.1b2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
81.7.1c1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1c2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.1d1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.1d2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1e1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1e2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.3a1	C	$3$	$3$	$3$	$0$	$0$	$0$	$0$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3a2	C	$3$	$3$	$3$	$0$	$0$	$0$	$0$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3b1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$0$	$0$	$1-{\zeta }_3$	$2+{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3b2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$0$	$0$	$2+{\zeta }_3$	$1-{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3c1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$1-{\zeta }_3$	$2+{\zeta }_3$	$0$	$0$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3c2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$2+{\zeta }_3$	$1-{\zeta }_3$	$0$	$0$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3d1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$2+{\zeta }_3$	$1-{\zeta }_3$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$0$	$0$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3d2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$1-{\zeta }_3$	$2+{\zeta }_3$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$0$	$0$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$