Galois Group: 9T20

• Label: 9T20
• Order: \(162\)
• n: \(9\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $C_3 \wr S_3 $

Group action invariants

Degree $n$ :		$9$
Transitive number $t$ :		$20$
Group :		$C_3 \wr S_3 $
CHM label :		$[3^{3}]S(3)=3wrS(3)$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,2,9), (3,6)(4,7)(5,8), (1,4,7)(2,5,8)(3,6,9)
$|\Aut(F/K)|$:		$3$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$
3:	$C_3$
6:	$S_3$, $C_6$
18:	$S_3\times C_3$
54:	$C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

9T20 x 2, 18T86 x 3, 27T37, 27T50 x 3, 27T70

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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 $	$6$	$3$	$(3,4,5)(6,8,7)$
$ 3, 3, 1, 1, 1 $	$3$	$3$	$(3,5,4)(6,8,7)$
$ 2, 2, 2, 1, 1, 1 $	$9$	$2$	$(3,6)(4,7)(5,8)$
$ 6, 1, 1, 1 $	$9$	$6$	$(3,6,4,7,5,8)$
$ 6, 1, 1, 1 $	$9$	$6$	$(3,6,5,8,4,7)$
$ 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, 2, 2, 2 $	$9$	$6$	$(1,2,9)(3,6)(4,7)(5,8)$
$ 6, 3 $	$9$	$6$	$(1,2,9)(3,6,4,7,5,8)$
$ 6, 3 $	$9$	$6$	$(1,2,9)(3,6,5,8,4,7)$
$ 3, 2, 2, 2 $	$9$	$6$	$(1,3)(2,4)(5,9)(6,8,7)$
$ 6, 3 $	$9$	$6$	$(1,3,2,4,9,5)(6,8,7)$
$ 3, 3, 3 $	$18$	$3$	$(1,3,6)(2,4,7)(5,8,9)$
$ 9 $	$18$	$9$	$(1,3,6,2,4,7,9,5,8)$
$ 9 $	$18$	$9$	$(1,3,6,9,5,8,2,4,7)$
$ 6, 3 $	$9$	$6$	$(1,3,9,5,2,4)(6,8,7)$
$ 3, 3, 3 $	$1$	$3$	$(1,9,2)(3,5,4)(6,8,7)$

Group invariants

Order:		$162=2 \cdot 3^{4}$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[162, 10]

Character table: Data not available.