Galois group: 9T19

• Label: 9T19
• Degree: $9$
• Order: $144$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $(C_3^2:C_8):C_2$

Group invariants

Abstract group:		$(C_3^2:C_8):C_2$
Order:		$144=2^{4} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$9$
Transitive number $t$:		$19$
CHM label:		$E(9):2D_{8}$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,6,4,5,2,3,8,7)$, $(1,2)(3,5)(6,7)$, $(1,2,9)(3,4,5)(6,7,8)$, $(1,4,7)(2,5,8)(3,6,9)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$
$16$:	$QD_{16}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

12T84, 18T68, 18T71, 18T73, 24T278, 24T279, 24T280, 36T171, 36T172, 36T175

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	Index	Representative
1A	$1^{9}$	$1$	$1$	$0$	$()$
2A	$2^{4},1$	$9$	$2$	$4$	$(2,9)(3,8)(4,7)(5,6)$
2B	$2^{3},1^{3}$	$12$	$2$	$3$	$(1,6)(2,7)(8,9)$
3A	$3^{3}$	$8$	$3$	$6$	$(1,2,9)(3,4,5)(6,7,8)$
4A	$4^{2},1$	$18$	$4$	$6$	$(2,5,9,6)(3,4,8,7)$
4B	$4^{2},1$	$36$	$4$	$6$	$(1,6,9,4)(3,8,7,5)$
6A	$6,3$	$24$	$6$	$7$	$(1,8,2,6,9,7)(3,5,4)$
8A1	$8,1$	$18$	$8$	$7$	$(2,7,5,3,9,4,6,8)$
8A-1	$8,1$	$18$	$8$	$7$	$(2,8,6,4,9,3,5,7)$

Malle's constant $a(G)$:     $1/3$

Character table

		1A	2A	2B	3A	4A	4B	6A	8A1	8A-1
	Size	1	9	12	8	18	36	24	18	18
	2 P	1A	1A	1A	3A	2A	2A	3A	4A	4A
	3 P	1A	2A	2B	1A	4A	4B	2B	8A1	8A-1
	Type
144.182.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
144.182.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$
144.182.1c	R	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
144.182.1d	R	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
144.182.2a	R	$2$	$2$	$0$	$2$	$-2$	$0$	$0$	$0$	$0$
144.182.2b1	C	$2$	$-2$	$0$	$2$	$0$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$
144.182.2b2	C	$2$	$-2$	$0$	$2$	$0$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$
144.182.8a	R	$8$	$0$	$2$	$-1$	$0$	$0$	$-1$	$0$	$0$
144.182.8b	R	$8$	$0$	$-2$	$-1$	$0$	$0$	$1$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$\left(t^{2} + 8\right) x^{9} + \left(-2 t^{3} - t^{2} - 16 t - 8\right) x^{8} + \left(t^{4} + 8 t^{2}\right) x^{7} + \left(t^{4} + 8 t^{2}\right) x^{6} + \left(t^{4} + 8 t^{2}\right) x^{5} + \left(t^{4} + 8 t^{2}\right) x^{4} + \left(t^{4} + 8 t^{2}\right) x^{3} + \left(t^{4} + 8 t^{2}\right) x^{2} + \left(t^{4} - t^{2}\right) x + \left(t^{4} + 2 t^{3} + t^{2}\right)$