Galois group: 9T6

• Label: 9T6
• Degree: $9$
• Order: $27$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_9:C_3$

Group invariants

Abstract group:		$C_9:C_3$
Order:		$27=3^{3}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

27T5

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	$3^{3}$	$1$	$3$	$6$	$(1,4,7)(2,5,8)(3,6,9)$
3A-1	$3^{3}$	$1$	$3$	$6$	$(1,7,4)(2,8,5)(3,9,6)$
3B1	$3^{2},1^{3}$	$3$	$3$	$4$	$(1,7,4)(2,5,8)$
3B-1	$3^{2},1^{3}$	$3$	$3$	$4$	$(1,4,7)(2,8,5)$
9A1	$9$	$3$	$9$	$8$	$(1,5,6,4,8,9,7,2,3)$
9A-1	$9$	$3$	$9$	$8$	$(1,6,8,7,3,5,4,9,2)$
9B1	$9$	$3$	$9$	$8$	$(1,2,6,4,5,9,7,8,3)$
9B-1	$9$	$3$	$9$	$8$	$(1,9,2,7,6,8,4,3,5)$
9C1	$9$	$3$	$9$	$8$	$(1,3,5,7,9,2,4,6,8)$
9C-1	$9$	$3$	$9$	$8$	$(1,8,6,4,2,9,7,5,3)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	3A1	3A-1	3B1	3B-1	9A1	9A-1	9B1	9B-1	9C1	9C-1
	Size	1	1	1	3	3	3	3	3	3	3	3
	3 P	1A	3A-1	3A1	3B-1	3B1	9A-1	9A1	9B-1	9B1	9C-1	9C1
	Type
27.4.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
27.4.1b1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
27.4.1b2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
27.4.1c1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
27.4.1c2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
27.4.1d1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
27.4.1d2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
27.4.1e1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
27.4.1e2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
27.4.3a1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
27.4.3a2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$\left(60466176 t^{8} - 26873856 t^{6} + 4478976 t^{4} - 331776 t^{2} + 9216\right) x^{9} + \left(-41150592 t^{10} - 112627584 t^{8} + 30751488 t^{6} - 4955904 t^{4} + 611712 t^{2} - 31104\right) x^{7} + \left(-22861440 t^{10} - 62570880 t^{8} + 17084160 t^{6} - 2753280 t^{4} + 339840 t^{2} - 17280\right) x^{6} + \left(7001316 t^{12} + 38007144 t^{10} + 50494428 t^{8} - 808272 t^{6} + 2196828 t^{4} - 36504 t^{2} + 23652\right) x^{5} + \left(7779240 t^{12} + 46040400 t^{10} + 66533400 t^{8} - 3745440 t^{6} + 2899800 t^{4} - 97200 t^{2} + 29160\right) x^{4} + \left(3306177 t^{12} + 19567170 t^{10} + 28276695 t^{8} - 1591812 t^{6} + 1232415 t^{4} - 41310 t^{2} + 12393\right) x^{3} + \left(669879 t^{12} + 3964590 t^{10} + 5729265 t^{8} - 322524 t^{6} + 249705 t^{4} - 8370 t^{2} + 2511\right) x^{2} + \left(64827 t^{12} + 383670 t^{10} + 554445 t^{8} - 31212 t^{6} + 24165 t^{4} - 810 t^{2} + 243\right) x + \left(2401 t^{12} + 14210 t^{10} + 20535 t^{8} - 1156 t^{6} + 895 t^{4} - 30 t^{2} + 9\right)$