Galois group: 36T7

• Label: 36T7
• Degree: $36$
• Order: $36$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3^2:C_4$

Group action invariants

Degree $n$:		$36$
Transitive number $t$:		$7$
Group:		$C_3^2:C_4$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$36$
Generators:		(1,32,2,31)(3,30,4,29)(5,28,6,27)(7,26,8,25)(9,11,10,12)(13,17,14,18)(15,19,16,20)(21,35,22,36)(23,34,24,33), (1,3,2,4)(5,34,6,33)(7,35,8,36)(9,31,10,32)(11,30,12,29)(13,27,14,28)(15,26,16,25)(17,24,18,23)(19,21,20,22), (1,14,2,13)(3,15,4,16)(5,9,6,10)(7,12,8,11)(17,35,18,36)(19,34,20,33)(21,32,22,31)(23,29,24,30)(25,27,26,28)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$6$:	$S_3$ x 4
$12$:	$C_3 : C_4$ x 4
$18$:	$C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 4: $C_4$

Degree 6: $S_3$ x 4

Degree 9: $C_3^2:C_2$

Degree 12: $C_3 : C_4$ x 4

Degree 18: $C_3^2 : C_2$

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	$1^{36}$	$1$	$1$	$0$	$()$
2A	$2^{18}$	$1$	$2$	$18$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1, 8,33)( 2, 7,34)( 3, 5,36)( 4, 6,35)( 9,15,17)(10,16,18)(11,14,19)(12,13,20)(21,28,30)(22,27,29)(23,25,32)(24,26,31)$
3B	$3^{12}$	$2$	$3$	$24$	$( 1,28,15)( 2,27,16)( 3,26,13)( 4,25,14)( 5,31,20)( 6,32,19)( 7,29,18)( 8,30,17)( 9,33,21)(10,34,22)(11,35,23)(12,36,24)$
3C	$3^{12}$	$2$	$3$	$24$	$( 1,21,17)( 2,22,18)( 3,24,20)( 4,23,19)( 5,26,12)( 6,25,11)( 7,27,10)( 8,28, 9)(13,36,31)(14,35,32)(15,33,30)(16,34,29)$
3D	$3^{12}$	$2$	$3$	$24$	$( 1, 9,30)( 2,10,29)( 3,12,31)( 4,11,32)( 5,13,24)( 6,14,23)( 7,16,22)( 8,15,21)(17,28,33)(18,27,34)(19,25,35)(20,26,36)$
4A1	$4^{9}$	$9$	$4$	$27$	$( 1,32, 2,31)( 3,30, 4,29)( 5,28, 6,27)( 7,26, 8,25)( 9,11,10,12)(13,17,14,18)(15,19,16,20)(21,35,22,36)(23,34,24,33)$
4A-1	$4^{9}$	$9$	$4$	$27$	$( 1,31, 2,32)( 3,29, 4,30)( 5,27, 6,28)( 7,25, 8,26)( 9,12,10,11)(13,18,14,17)(15,20,16,19)(21,36,22,35)(23,33,24,34)$
6A	$6^{6}$	$2$	$6$	$30$	$( 1,22,17, 2,21,18)( 3,23,20, 4,24,19)( 5,25,12, 6,26,11)( 7,28,10, 8,27, 9)(13,35,31,14,36,32)(15,34,30,16,33,29)$
6B	$6^{6}$	$2$	$6$	$30$	$( 1,10,30, 2, 9,29)( 3,11,31, 4,12,32)( 5,14,24, 6,13,23)( 7,15,22, 8,16,21)(17,27,33,18,28,34)(19,26,35,20,25,36)$
6C	$6^{6}$	$2$	$6$	$30$	$( 1, 7,33, 2, 8,34)( 3, 6,36, 4, 5,35)( 9,16,17,10,15,18)(11,13,19,12,14,20)(21,27,30,22,28,29)(23,26,32,24,25,31)$
6D	$6^{6}$	$2$	$6$	$30$	$( 1,27,15, 2,28,16)( 3,25,13, 4,26,14)( 5,32,20, 6,31,19)( 7,30,18, 8,29,17)( 9,34,21,10,33,22)(11,36,23,12,35,24)$

Malle's constant $a(G)$:     $1/18$

Group invariants

Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		36.7
Character table:

		1A	2A	3A	3B	3C	3D	4A1	4A-1	6A	6B	6C	6D
	Size	1	1	2	2	2	2	9	9	2	2	2	2
	2 P	1A	1A	3C	3D	3A	3B	2A	2A	3A	3B	3C	3D
	3 P	1A	2A	1A	1A	1A	1A	4A-1	4A1	2A	2A	2A	2A
	Type
36.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
36.7.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
36.7.1c1	C	$1$	$-1$	$1$	$1$	$1$	$1$	$-i$	$i$	$-1$	$-1$	$-1$	$-1$
36.7.1c2	C	$1$	$-1$	$1$	$1$	$1$	$1$	$i$	$-i$	$-1$	$-1$	$-1$	$-1$
36.7.2a	R	$2$	$2$	$-1$	$-1$	$-1$	$2$	$0$	$0$	$-1$	$-1$	$-1$	$2$
36.7.2b	R	$2$	$2$	$-1$	$-1$	$2$	$-1$	$0$	$0$	$-1$	$-1$	$2$	$-1$
36.7.2c	R	$2$	$2$	$-1$	$2$	$-1$	$-1$	$0$	$0$	$-1$	$2$	$-1$	$-1$
36.7.2d	R	$2$	$2$	$2$	$-1$	$-1$	$-1$	$0$	$0$	$2$	$-1$	$-1$	$-1$
36.7.2e	S	$2$	$-2$	$-1$	$-1$	$-1$	$2$	$0$	$0$	$1$	$1$	$1$	$-2$
36.7.2f	S	$2$	$-2$	$-1$	$-1$	$2$	$-1$	$0$	$0$	$1$	$1$	$-2$	$1$
36.7.2g	S	$2$	$-2$	$-1$	$2$	$-1$	$-1$	$0$	$0$	$1$	$-2$	$1$	$1$
36.7.2h	S	$2$	$-2$	$2$	$-1$	$-1$	$-1$	$0$	$0$	$-2$	$1$	$1$	$1$