Galois group: 9T17

• Label: 9T17
• Degree: $9$
• Order: $81$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_3 \wr C_3 $

Group invariants

Abstract group:		$C_3 \wr C_3 $
Order:		$81=3^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$9$
Transitive number $t$:		$17$
CHM label:		$[3^{3}]3=3wr3$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		$(1,2,9)$, $(1,4,7)(2,5,8)(3,6,9)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$3$:	$C_3$ x 4
$9$:	$C_3^2$
$27$:	$C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

9T17 x 2, 27T19, 27T21, 27T27 x 3

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	$1^{9}$	$1$	$1$	$0$	$()$
3A1	$3^{3}$	$1$	$3$	$6$	$(1,9,2)(3,5,4)(6,8,7)$
3A-1	$3^{3}$	$1$	$3$	$6$	$(1,2,9)(3,4,5)(6,7,8)$
3B1	$3^{2},1^{3}$	$3$	$3$	$4$	$(3,5,4)(6,7,8)$
3B-1	$3^{2},1^{3}$	$3$	$3$	$4$	$(3,4,5)(6,8,7)$
3C1	$3,1^{6}$	$3$	$3$	$2$	$(1,9,2)$
3C-1	$3,1^{6}$	$3$	$3$	$2$	$(1,2,9)$
3D1	$3^{3}$	$3$	$3$	$6$	$(1,2,9)(3,5,4)(6,8,7)$
3D-1	$3^{3}$	$3$	$3$	$6$	$(1,2,9)(3,5,4)(6,7,8)$
3E1	$3^{2},1^{3}$	$3$	$3$	$4$	$(3,4,5)(6,7,8)$
3E-1	$3^{2},1^{3}$	$3$	$3$	$4$	$(3,5,4)(6,8,7)$
3F1	$3^{3}$	$9$	$3$	$6$	$(1,8,4)(2,6,5)(3,9,7)$
3F-1	$3^{3}$	$9$	$3$	$6$	$(1,4,7)(2,5,8)(3,6,9)$
9A1	$9$	$9$	$9$	$8$	$(1,3,6,9,5,8,2,4,7)$
9A-1	$9$	$9$	$9$	$8$	$(1,6,5,2,7,3,9,8,4)$
9B1	$9$	$9$	$9$	$8$	$(1,5,8,2,3,6,9,4,7)$
9B-1	$9$	$9$	$9$	$8$	$(1,7,3,9,6,5,2,8,4)$

Malle's constant $a(G)$:     $1/2$

Character table

		1A	3A1	3A-1	3B1	3B-1	3C1	3C-1	3D1	3D-1	3E1	3E-1	3F1	3F-1	9A1	9A-1	9B1	9B-1
	Size	1	1	1	3	3	3	3	3	3	3	3	9	9	9	9	9	9
	3 P	1A	3A-1	3A1	3B-1	3B1	3C-1	3C1	3D-1	3D1	3E-1	3E1	3F-1	3F1	9A-1	9A1	9B-1	9B1
	Type
81.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
81.7.1b1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
81.7.1b2	C	$1$	$1$	$1$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
81.7.1c1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.1c2	C	$1$	$1$	$1$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1d1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1d2	C	$1$	$1$	$1$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.1e1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1e2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.3a1	C	$3$	$3$	$3$	$0$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3a2	C	$3$	$3$	$3$	$0$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3b1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$-2-{\zeta }_3$	$0$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$1-{\zeta }_3$	$-1+{\zeta }_3$	$0$	$2+{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3b2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$-1+{\zeta }_3$	$0$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$2+{\zeta }_3$	$-2-{\zeta }_3$	$0$	$1-{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3c1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$1+2{\zeta }_3$	$0$	$1-{\zeta }_3$	$2+{\zeta }_3$	$-2-{\zeta }_3$	$-1-2{\zeta }_3$	$0$	$-1+{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3c2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$-1-2{\zeta }_3$	$0$	$2+{\zeta }_3$	$1-{\zeta }_3$	$-1+{\zeta }_3$	$1+2{\zeta }_3$	$0$	$-2-{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3d1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$1-{\zeta }_3$	$0$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$1+2{\zeta }_3$	$2+{\zeta }_3$	$0$	$-1-2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3d2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$2+{\zeta }_3$	$0$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$-1-2{\zeta }_3$	$1-{\zeta }_3$	$0$	$1+2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$x^{9} + t x^{8} - 36 x^{7} - 20 t x^{6} + 342 x^{5} + 102 t x^{4} - 900 x^{3} - 84 t x^{2} + 81 x + t$