Galois Group: 9T6

• Label: 9T6
• Order: \(27\)
• n: \(9\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: Yes
• Group: $C_9:C_3$

Group action invariants

Degree $n$ :		$9$
Transitive number $t$ :		$6$
Group :		$C_9:C_3$
CHM label :		$1/3[3^{3}]3$
Parity:		$1$
Primitive:		No
Nilpotency class:		$2$
Generators:		(1,4,7)(2,8,5), (1,2,3,4,5,6,7,8,9)
$|\Aut(F/K)|$:		$3$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

27T5

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

Group invariants

Order:		$27=3^{3}$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[27, 4]

Character table:

3 3 2 2 2 2 2 2 2 2 3 3 1a 3a 3b 9a 9b 9c 9d 9e 9f 3c 3d 2P 1a 3b 3a 9e 9d 9f 9b 9a 9c 3d 3c 3P 1a 1a 1a 3c 3c 3c 3d 3d 3d 1a 1a 5P 1a 3b 3a 9e 9d 9f 9b 9a 9c 3d 3c 7P 1a 3a 3b 9a 9b 9c 9d 9e 9f 3c 3d X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 A A A /A /A /A 1 1 X.3 1 1 1 /A /A /A A A A 1 1 X.4 1 A /A 1 A /A /A 1 A 1 1 X.5 1 /A A 1 /A A A 1 /A 1 1 X.6 1 A /A A /A 1 A /A 1 1 1 X.7 1 /A A /A A 1 /A A 1 1 1 X.8 1 A /A /A 1 A 1 A /A 1 1 X.9 1 /A A A 1 /A 1 /A A 1 1 X.10 3 . . . . . . . . B /B X.11 3 . . . . . . . . /B B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3