Galois group: 37T6

• Label: 37T6
• Degree: $37$
• Order: $333$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{37}:C_{9}$

Group action invariants

Degree $n$:		$37$
Transitive number $t$:		$6$
Group:		$C_{37}:C_{9}$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (1,16,34,26,9,33,10,12,7)(2,32,31,15,18,29,20,24,14)(3,11,28,4,27,25,30,36,21)(5,6,22,19,8,17,13,23,35)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$3$:	$C_3$
$9$:	$C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

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	$1^{37}$	$1$	$1$	$0$	$()$
3A1	$3^{12},1$	$37$	$3$	$24$	$( 2,11,27)( 3,21,16)( 4,31, 5)( 6,14,20)( 7,24, 9)( 8,34,35)(10,17,13)(12,37,28)(15,30,32)(18,23,36)(19,33,25)(22,26,29)$
3A-1	$3^{12},1$	$37$	$3$	$24$	$( 2,27,11)( 3,16,21)( 4, 5,31)( 6,20,14)( 7, 9,24)( 8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)$
9A1	$9^{4},1$	$37$	$9$	$32$	$( 2,17,35,27,10,34,11,13, 8)( 3,33,32,16,19,30,21,25,15)( 4,12,29, 5,28,26,31,37,22)( 6, 7,23,20, 9,18,14,24,36)$
9A-1	$9^{4},1$	$37$	$9$	$32$	$( 2,13,34,27,17, 8,11,10,35)( 3,25,30,16,33,15,21,19,32)( 4,37,26, 5,12,22,31,28,29)( 6,24,18,20, 7,36,14, 9,23)$
9A2	$9^{4},1$	$37$	$9$	$32$	$( 2,10, 8,27,13,35,11,17,34)( 3,19,15,16,25,32,21,33,30)( 4,28,22, 5,37,29,31,12,26)( 6, 9,36,20,24,23,14, 7,18)$
9A-2	$9^{4},1$	$37$	$9$	$32$	$( 2,35,10,11, 8,17,27,34,13)( 3,32,19,21,15,33,16,30,25)( 4,29,28,31,22,12, 5,26,37)( 6,23, 9,14,36, 7,20,18,24)$
9A4	$9^{4},1$	$37$	$9$	$32$	$( 2, 8,13,11,34,10,27,35,17)( 3,15,25,21,30,19,16,32,33)( 4,22,37,31,26,28, 5,29,12)( 6,36,24,14,18, 9,20,23, 7)$
9A-4	$9^{4},1$	$37$	$9$	$32$	$( 2,34,17,11,35,13,27, 8,10)( 3,30,33,21,32,25,16,15,19)( 4,26,12,31,29,37, 5,22,28)( 6,18, 7,14,23,24,20,36, 9)$
37A1	$37$	$9$	$37$	$36$	$( 1,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
37A-1	$37$	$9$	$37$	$36$	$( 1,35,32,29,26,23,20,17,14,11, 8, 5, 2,36,33,30,27,24,21,18,15,12, 9, 6, 3,37,34,31,28,25,22,19,16,13,10, 7, 4)$
37A2	$37$	$9$	$37$	$36$	$( 1,36,34,32,30,28,26,24,22,20,18,16,14,12,10, 8, 6, 4, 2,37,35,33,31,29,27,25,23,21,19,17,15,13,11, 9, 7, 5, 3)$
37A-2	$37$	$9$	$37$	$36$	$( 1,33,28,23,18,13, 8, 3,35,30,25,20,15,10, 5,37,32,27,22,17,12, 7, 2,34,29,24,19,14, 9, 4,36,31,26,21,16,11, 6)$

Malle's constant $a(G)$:     $1/24$

Group invariants

Order:		$333=3^{2} \cdot 37$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		333.3
Character table:

		1A	3A1	3A-1	9A1	9A-1	9A2	9A-2	9A4	9A-4	37A1	37A-1	37A2	37A-2
	Size	1	37	37	37	37	37	37	37	37	9	9	9	9
	3 P	1A	3A-1	3A1	9A1	9A-2	9A4	9A2	9A-1	9A-4	37A2	37A-2	37A-1	37A1
	37 P	1A	1A	1A	3A-1	3A-1	3A-1	3A1	3A1	3A1	37A-1	37A1	37A-2	37A2
	Type
333.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
333.3.1b1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$
333.3.1b2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$
333.3.1c1	C	$1$	${\zeta }_9^{-3}$	${\zeta }_9^3$	${\zeta }_9^{-4}$	${\zeta }_9^4$	${\zeta }_9$	${\zeta }_9^{-1}$	${\zeta }_9^2$	${\zeta }_9^{-2}$	$1$	$1$	$1$	$1$
333.3.1c2	C	$1$	${\zeta }_9^3$	${\zeta }_9^{-3}$	${\zeta }_9^4$	${\zeta }_9^{-4}$	${\zeta }_9^{-1}$	${\zeta }_9$	${\zeta }_9^{-2}$	${\zeta }_9^2$	$1$	$1$	$1$	$1$
333.3.1c3	C	$1$	${\zeta }_9^{-3}$	${\zeta }_9^3$	${\zeta }_9^2$	${\zeta }_9^{-2}$	${\zeta }_9^4$	${\zeta }_9^{-4}$	${\zeta }_9^{-1}$	${\zeta }_9$	$1$	$1$	$1$	$1$
333.3.1c4	C	$1$	${\zeta }_9^3$	${\zeta }_9^{-3}$	${\zeta }_9^{-2}$	${\zeta }_9^2$	${\zeta }_9^{-4}$	${\zeta }_9^4$	${\zeta }_9$	${\zeta }_9^{-1}$	$1$	$1$	$1$	$1$
333.3.1c5	C	$1$	${\zeta }_9^{-3}$	${\zeta }_9^3$	${\zeta }_9^{-1}$	${\zeta }_9$	${\zeta }_9^{-2}$	${\zeta }_9^2$	${\zeta }_9^{-4}$	${\zeta }_9^4$	$1$	$1$	$1$	$1$
333.3.1c6	C	$1$	${\zeta }_9^3$	${\zeta }_9^{-3}$	${\zeta }_9$	${\zeta }_9^{-1}$	${\zeta }_9^2$	${\zeta }_9^{-2}$	${\zeta }_9^4$	${\zeta }_9^{-4}$	$1$	$1$	$1$	$1$
333.3.9a1	C	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-5}+{\zeta }_{37}^2+{\zeta }_{37}^{14}+{\zeta }_{37}^{15}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-14}+{\zeta }_{37}^{-2}+{\zeta }_{37}^5+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{13}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-1}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}+{\zeta }_{37}^7+{\zeta }_{37}^9+{\zeta }_{37}^{10}+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$
333.3.9a2	C	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-14}+{\zeta }_{37}^{-2}+{\zeta }_{37}^5+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{13}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-5}+{\zeta }_{37}^2+{\zeta }_{37}^{14}+{\zeta }_{37}^{15}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}+{\zeta }_{37}^7+{\zeta }_{37}^9+{\zeta }_{37}^{10}+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-1}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^{11}$
333.3.9a3	C	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-1}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}+{\zeta }_{37}^7+{\zeta }_{37}^9+{\zeta }_{37}^{10}+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-14}+{\zeta }_{37}^{-2}+{\zeta }_{37}^5+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{13}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-5}+{\zeta }_{37}^2+{\zeta }_{37}^{14}+{\zeta }_{37}^{15}+{\zeta }_{37}^{18}$
333.3.9a4	C	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}+{\zeta }_{37}^7+{\zeta }_{37}^9+{\zeta }_{37}^{10}+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-1}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-5}+{\zeta }_{37}^2+{\zeta }_{37}^{14}+{\zeta }_{37}^{15}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-14}+{\zeta }_{37}^{-2}+{\zeta }_{37}^5+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{13}+{\zeta }_{37}^{17}$