Galois group: 36T9

• Label: 36T9
• Degree: $36$
• Order: $36$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_9:C_4$

Group action invariants

Degree $n$:		$36$
Transitive number $t$:		$9$
Group:		$C_9:C_4$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$36$
Generators:		(1,3,2,4)(5,34,6,33)(7,35,8,36)(9,32,10,31)(11,30,12,29)(13,28,14,27)(15,26,16,25)(17,23,18,24)(19,21,20,22), (1,34,32,28,24,19,16,12,7)(2,33,31,27,23,20,15,11,8)(3,35,29,25,21,17,14,10,6)(4,36,30,26,22,18,13,9,5)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$6$:	$S_3$
$12$:	$C_3 : C_4$
$18$:	$D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $S_3$

Degree 9: $D_{9}$

Degree 12: $C_3 : C_4$

Degree 18: $D_9$

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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$9$	$4$	$( 1, 3, 2, 4)( 5,34, 6,33)( 7,35, 8,36)( 9,32,10,31)(11,30,12,29)(13,28,14,27) (15,26,16,25)(17,23,18,24)(19,21,20,22)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$9$	$4$	$( 1, 4, 2, 3)( 5,33, 6,34)( 7,36, 8,35)( 9,31,10,32)(11,29,12,30)(13,27,14,28) (15,25,16,26)(17,24,18,23)(19,22,20,21)$
	$ 9, 9, 9, 9 $	$2$	$9$	$( 1, 7,12,16,19,24,28,32,34)( 2, 8,11,15,20,23,27,31,33)( 3, 6,10,14,17,21,25, 29,35)( 4, 5, 9,13,18,22,26,30,36)$
	$ 18, 18 $	$2$	$18$	$( 1, 8,12,15,19,23,28,31,34, 2, 7,11,16,20,24,27,32,33)( 3, 5,10,13,17,22,25, 30,35, 4, 6, 9,14,18,21,26,29,36)$
	$ 18, 18 $	$2$	$18$	$( 1,11,19,27,34, 8,16,23,32, 2,12,20,28,33, 7,15,24,31)( 3, 9,17,26,35, 5,14, 22,29, 4,10,18,25,36, 6,13,21,30)$
	$ 9, 9, 9, 9 $	$2$	$9$	$( 1,12,19,28,34, 7,16,24,32)( 2,11,20,27,33, 8,15,23,31)( 3,10,17,25,35, 6,14, 21,29)( 4, 9,18,26,36, 5,13,22,30)$
	$ 6, 6, 6, 6, 6, 6 $	$2$	$6$	$( 1,15,28, 2,16,27)( 3,13,25, 4,14,26)( 5,17,30, 6,18,29)( 7,20,32, 8,19,31) ( 9,21,36,10,22,35)(11,24,33,12,23,34)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1,16,28)( 2,15,27)( 3,14,25)( 4,13,26)( 5,18,30)( 6,17,29)( 7,19,32) ( 8,20,31)( 9,22,36)(10,21,35)(11,23,33)(12,24,34)$
	$ 9, 9, 9, 9 $	$2$	$9$	$( 1,19,34,16,32,12,28, 7,24)( 2,20,33,15,31,11,27, 8,23)( 3,17,35,14,29,10,25, 6,21)( 4,18,36,13,30, 9,26, 5,22)$
	$ 18, 18 $	$2$	$18$	$( 1,20,34,15,32,11,28, 8,24, 2,19,33,16,31,12,27, 7,23)( 3,18,35,13,29, 9,25, 5,21, 4,17,36,14,30,10,26, 6,22)$

Group invariants

Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		36.1
Character table:

		1A	2A	3A	4A1	4A-1	6A	9A1	9A2	9A4	18A1	18A5	18A7
	Size	1	1	2	9	9	2	2	2	2	2	2	2
	2 P	1A	1A	3A	2A	2A	3A	9A2	9A4	9A1	9A4	9A2	9A1
	3 P	1A	2A	1A	4A-1	4A1	2A	3A	3A	3A	6A	6A	6A
	Type
36.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
36.1.1b	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
36.1.1c1	C	$1$	$-1$	$1$	$-i$	$i$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
36.1.1c2	C	$1$	$-1$	$1$	$i$	$-i$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
36.1.2a	R	$2$	$2$	$2$	$0$	$0$	$2$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
36.1.2b	S	$2$	$-2$	$2$	$0$	$0$	$-2$	$-1$	$-1$	$-1$	$1$	$1$	$1$
36.1.2c1	R	$2$	$2$	$-1$	$0$	$0$	$-1$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-4}+{\zeta }_9^4$
36.1.2c2	R	$2$	$2$	$-1$	$0$	$0$	$-1$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-2}+{\zeta }_9^2$
36.1.2c3	R	$2$	$2$	$-1$	$0$	$0$	$-1$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-1}+{\zeta }_9$
36.1.2d1	S	$2$	$-2$	$-1$	$0$	$0$	$1$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-2}+{\zeta }_9^2$	$-{\zeta }_9^{-2}-{\zeta }_9^2$	$-{\zeta }_9^{-1}-{\zeta }_9$	$-{\zeta }_9^{-4}-{\zeta }_9^4$
36.1.2d2	S	$2$	$-2$	$-1$	$0$	$0$	$1$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-1}+{\zeta }_9$	$-{\zeta }_9^{-1}-{\zeta }_9$	$-{\zeta }_9^{-4}-{\zeta }_9^4$	$-{\zeta }_9^{-2}-{\zeta }_9^2$
36.1.2d3	S	$2$	$-2$	$-1$	$0$	$0$	$1$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-4}+{\zeta }_9^4$	$-{\zeta }_9^{-4}-{\zeta }_9^4$	$-{\zeta }_9^{-2}-{\zeta }_9^2$	$-{\zeta }_9^{-1}-{\zeta }_9$