Galois group: 9T14

• Label: 9T14
• Degree: $9$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_3^2:Q_8$

Group action invariants

Degree $n$:		$9$
Transitive number $t$:		$14$
Group:		$C_3^2:Q_8$
CHM label:		$M(9)=E(9):Q_{8}$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,9)(3,4,5)(6,7,8), (1,8,2,4)(3,5,6,7), (1,6,2,3)(4,7,8,5), (1,4,7)(2,5,8)(3,6,9)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

12T47, 18T35 x 3, 24T82, 36T55

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

Group invariants

Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		72.41
Character table:

		1A	2A	3A	4A	4B	4C
	Size	1	9	8	18	18	18
	2 P	1A	1A	3A	2A	2A	2A
	3 P	1A	2A	1A	4A	4B	4C
	Type
72.41.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
72.41.1b	R	$1$	$1$	$1$	$-1$	$-1$	$1$
72.41.1c	R	$1$	$1$	$1$	$-1$	$1$	$-1$
72.41.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$
72.41.2a	S	$2$	$-2$	$2$	$0$	$0$	$0$
72.41.8a	R	$8$	$0$	$-1$	$0$	$0$	$0$