Galois group: 9T2

• Label: 9T2
• Degree: $9$
• Order: $9$
• Cyclic: no
• Abelian: yes
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_3^2$

Group invariants

Abstract group:		$C_3^2$
Order:		$9=3^{2}$
Cyclic:		no
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$3$:	$C_3$ x 4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Low degree siblings

There are no siblings 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	$1^{9}$	$1$	$1$	$0$	$()$
3A1	$3^{3}$	$1$	$3$	$6$	$(1,2,9)(3,4,5)(6,7,8)$
3A-1	$3^{3}$	$1$	$3$	$6$	$(1,9,2)(3,5,4)(6,8,7)$
3B1	$3^{3}$	$1$	$3$	$6$	$(1,4,7)(2,5,8)(3,6,9)$
3B-1	$3^{3}$	$1$	$3$	$6$	$(1,7,4)(2,8,5)(3,9,6)$
3C1	$3^{3}$	$1$	$3$	$6$	$(1,5,6)(2,3,7)(4,8,9)$
3C-1	$3^{3}$	$1$	$3$	$6$	$(1,6,5)(2,7,3)(4,9,8)$
3D1	$3^{3}$	$1$	$3$	$6$	$(1,3,8)(2,4,6)(5,7,9)$
3D-1	$3^{3}$	$1$	$3$	$6$	$(1,8,3)(2,6,4)(5,9,7)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	3A1	3A-1	3B1	3B-1	3C1	3C-1	3D1	3D-1
	Size	1	1	1	1	1	1	1	1	1
	3 P	1A	3A-1	3A1	3B-1	3B1	3C-1	3C1	3D-1	3D1
	Type
9.2.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
9.2.1b1	C	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	$1$
9.2.1b2	C	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	$1$
9.2.1c1	C	$1$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$
9.2.1c2	C	$1$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$
9.2.1d1	C	$1$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$
9.2.1d2	C	$1$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$
9.2.1e1	C	$1$	$1$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
9.2.1e2	C	$1$	$1$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$

Regular extensions

$f_{ 1 } =$

$x^{9} + \left(-18 t^{2} - 15\right) x^{7} + 6 t x^{6} + \left(81 t^{4} + 135 t^{2} + 63\right) x^{5} + \left(-72 t^{3} - 60 t\right) x^{4} + \left(-90 t^{4} - 192 t^{2} - 85\right) x^{3} + 54 t x^{2} + \left(9 t^{4} + 75 t^{2} + 36\right) x + 8 t^{3}$