Galois group: 36T36

• Label: 36T36
• Degree: $36$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2\times C_3^2:C_4$

Group action invariants

Degree $n$:		$36$
Transitive number $t$:		$36$
Group:		$C_2\times C_3^2:C_4$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		(1,14,28,2,13,27)(3,15,26,4,16,25)(5,20,31,6,19,32)(7,18,29,8,17,30)(9,21,35,10,22,36)(11,23,34,12,24,33), (1,3)(2,4)(5,17,33,22)(6,18,34,21)(7,19,35,23)(8,20,36,24)(9,13,29,28)(10,14,30,27)(11,15,32,25)(12,16,31,26)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$C_4\times C_2$
$36$:	$C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $C_3^2:C_4$ x 2

Degree 9: $C_3^2:C_4$

Degree 12: 12T40 x 2

Degree 18: $C_3^2 : C_4$, 18T27 x 2

Low degree siblings

12T40 x 2, 12T41 x 2, 18T27 x 2, 24T76 x 2, 36T35

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	$1^{36}$	$1$	$1$	$0$	$()$
2A	$2^{18}$	$1$	$2$	$18$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B	$2^{16},1^{4}$	$9$	$2$	$16$	$( 5,33)( 6,34)( 7,35)( 8,36)( 9,29)(10,30)(11,32)(12,31)(13,28)(14,27)(15,25)(16,26)(17,22)(18,21)(19,23)(20,24)$
2C	$2^{18}$	$9$	$2$	$18$	$( 1, 2)( 3, 4)( 5,34)( 6,33)( 7,36)( 8,35)( 9,30)(10,29)(11,31)(12,32)(13,27)(14,28)(15,26)(16,25)(17,21)(18,22)(19,24)(20,23)$
3A	$3^{12}$	$4$	$3$	$24$	$( 1,19,23)( 2,20,24)( 3,17,22)( 4,18,21)( 5,12,28)( 6,11,27)( 7, 9,26)( 8,10,25)(13,31,33)(14,32,34)(15,30,36)(16,29,35)$
3B	$3^{12}$	$4$	$3$	$24$	$( 1,31,12)( 2,32,11)( 3,29, 9)( 4,30,10)( 5,23,13)( 6,24,14)( 7,22,16)( 8,21,15)(17,35,26)(18,36,25)(19,33,28)(20,34,27)$
4A1	$4^{8},2^{2}$	$9$	$4$	$26$	$( 1, 3)( 2, 4)( 5,22,33,17)( 6,21,34,18)( 7,23,35,19)( 8,24,36,20)( 9,28,29,13)(10,27,30,14)(11,25,32,15)(12,26,31,16)$
4A-1	$4^{8},2^{2}$	$9$	$4$	$26$	$( 1, 4)( 2, 3)( 5,18,33,21)( 6,17,34,22)( 7,20,35,24)( 8,19,36,23)( 9,14,29,27)(10,13,30,28)(11,16,32,26)(12,15,31,25)$
4B1	$4^{8},2^{2}$	$9$	$4$	$26$	$( 1, 4)( 2, 3)( 5,21,33,18)( 6,22,34,17)( 7,24,35,20)( 8,23,36,19)( 9,27,29,14)(10,28,30,13)(11,26,32,16)(12,25,31,15)$
4B-1	$4^{8},2^{2}$	$9$	$4$	$26$	$( 1, 3)( 2, 4)( 5,17,33,22)( 6,18,34,21)( 7,19,35,23)( 8,20,36,24)( 9,13,29,28)(10,14,30,27)(11,15,32,25)(12,16,31,26)$
6A	$6^{6}$	$4$	$6$	$30$	$( 1,20,23, 2,19,24)( 3,18,22, 4,17,21)( 5,11,28, 6,12,27)( 7,10,26, 8, 9,25)(13,32,33,14,31,34)(15,29,36,16,30,35)$
6B	$6^{6}$	$4$	$6$	$30$	$( 1,32,12, 2,31,11)( 3,30, 9, 4,29,10)( 5,24,13, 6,23,14)( 7,21,16, 8,22,15)(17,36,26,18,35,25)(19,34,28,20,33,27)$

Malle's constant $a(G)$:     $1/16$

Group invariants

Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		72.45
Character table:

		1A	2A	2B	2C	3A	3B	4A1	4A-1	4B1	4B-1	6A	6B
	Size	1	1	9	9	4	4	9	9	9	9	4	4
	2 P	1A	1A	1A	1A	3A	3B	2B	2B	2B	2B	3A	3B
	3 P	1A	2A	2B	2C	1A	1A	4B-1	4A-1	4A1	4B1	2A	2A
	Type
72.45.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.45.1b	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
72.45.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
72.45.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
72.45.1e1	C	$1$	$-1$	$-1$	$1$	$1$	$1$	$-i$	$i$	$-i$	$i$	$-1$	$-1$
72.45.1e2	C	$1$	$-1$	$-1$	$1$	$1$	$1$	$i$	$-i$	$i$	$-i$	$-1$	$-1$
72.45.1f1	C	$1$	$1$	$-1$	$-1$	$1$	$1$	$-i$	$i$	$i$	$-i$	$1$	$1$
72.45.1f2	C	$1$	$1$	$-1$	$-1$	$1$	$1$	$i$	$-i$	$-i$	$i$	$1$	$1$
72.45.4a	R	$4$	$4$	$0$	$0$	$-2$	$1$	$0$	$0$	$0$	$0$	$-2$	$1$
72.45.4b	R	$4$	$4$	$0$	$0$	$1$	$-2$	$0$	$0$	$0$	$0$	$1$	$-2$
72.45.4c	R	$4$	$-4$	$0$	$0$	$-2$	$1$	$0$	$0$	$0$	$0$	$2$	$-1$
72.45.4d	R	$4$	$-4$	$0$	$0$	$1$	$-2$	$0$	$0$	$0$	$0$	$-1$	$2$