Galois group: 36T35

• Label: 36T35
• 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 $t$:		$35$
Group:		$C_2\times C_3^2:C_4$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		(1,13,28,2,14,27)(3,15,25,4,16,26)(5,20,32,6,19,31)(7,17,30,8,18,29)(9,21,36,10,22,35)(11,24,33,12,23,34), (1,25,33,9)(2,26,34,10)(3,27,36,12)(4,28,35,11)(5,7,6,8)(13,17,24,30)(14,18,23,29)(15,19,21,31)(16,20,22,32)

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: $C_2$

Degree 3: None

Degree 4: $C_4$

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

Degree 9: $C_3^2:C_4$

Degree 12: 12T41 x 2

Degree 18: $C_3^2 : C_4$

Low degree siblings

12T40 x 2, 12T41 x 2, 18T27 x 2, 24T76 x 2, 36T36

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

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	$4$	$-4$	$0$	$0$	$1$	$-2$	$0$	$0$	$0$	$0$	$-1$	$2$