Galois group: 9T4

• Label: 9T4
• Degree: $9$
• Order: $18$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $S_3\times C_3$

Group invariants

Abstract group:		$S_3\times C_3$
Order:		$18=2 \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$9$
Transitive number $t$:		$4$
CHM label:		$S(3)[x]3$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		$(1,2,9)(3,4,5)(6,7,8)$, $(1,2)(4,5)(7,8)$, $(1,4,7)(2,5,8)(3,6,9)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$S_3$, $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Low degree siblings

6T5, 18T3

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	$2^{3},1^{3}$	$3$	$2$	$3$	$(2,9)(3,5)(6,8)$
3A1	$3^{3}$	$1$	$3$	$6$	$(1,4,7)(2,5,8)(3,6,9)$
3A-1	$3^{3}$	$1$	$3$	$6$	$(1,7,4)(2,8,5)(3,9,6)$
3B	$3^{3}$	$2$	$3$	$6$	$(1,2,9)(3,4,5)(6,7,8)$
3C1	$3^{3}$	$2$	$3$	$6$	$(1,5,6)(2,3,7)(4,8,9)$
3C-1	$3^{3}$	$2$	$3$	$6$	$(1,8,3)(2,6,4)(5,9,7)$
6A1	$6,3$	$3$	$6$	$7$	$(1,7,4)(2,6,5,9,8,3)$
6A-1	$6,3$	$3$	$6$	$7$	$(1,4,7)(2,3,8,9,5,6)$

Malle's constant $a(G)$:     $1/3$

Character table

		1A	2A	3A1	3A-1	3B	3C1	3C-1	6A1	6A-1
	Size	1	3	1	1	2	2	2	3	3
	2 P	1A	1A	3A-1	3A1	3B	3C-1	3C1	3A1	3A-1
	3 P	1A	2A	1A	1A	1A	1A	1A	2A	2A
	Type
18.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
18.3.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
18.3.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
18.3.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
18.3.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
18.3.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
18.3.2a	R	$2$	$0$	$2$	$2$	$-1$	$-1$	$-1$	$0$	$0$
18.3.2b1	C	$2$	$0$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$-1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$0$
18.3.2b2	C	$2$	$0$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$-1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$x^{9} + \left(t^{2} + t + 7\right) x^{7} + \left(t^{2} + t + 4\right) x^{6} + \left(2 t^{2} + 2 t + 14\right) x^{5} + \left(2 t^{2} + 2 t + 14\right) x^{4} + \left(t^{2} + t + 10\right) x^{3} + \left(t^{2} + t + 7\right) x^{2} - 1$