Galois group: 37T3

• Label: 37T3
• Degree: $37$
• Order: $111$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{37}:C_{3}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$3$:	$C_3$

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

Group invariants

Order:		$111=3 \cdot 37$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		111.1
Character table:

1A

3A1

3A-1

37A1

37A-1

37A2

37A-2

37A3

37A-3

37A5

37A-5

37A6

37A-6

37A9

37A-9

Size

1

37

37

3

3

3

3

3

3

3

3

3

3

3

3

3 P

1A

3A-1

3A1

37A2

37A3

37A-5

37A5

37A-3

37A-6

37A1

37A-9

37A9

37A-2

37A-1

37A6

37 P

1A

1A

1A

37A3

37A6

37A-1

37A1

37A-6

37A-9

37A2

37A5

37A-5

37A-3

37A-2

37A9

Type

111.1.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

111.1.1b1

C

$1$

${\zeta }_3^{-1}$

${\zeta }_3$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

111.1.1b2

C

$1$

${\zeta }_3$

${\zeta }_3^{-1}$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

111.1.3a1

C

$3$

$0$

$0$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

111.1.3a2

C

$3$

$0$

$0$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

111.1.3a3

C

$3$

$0$

$0$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

111.1.3a4

C

$3$

$0$

$0$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

111.1.3a5

C

$3$

$0$

$0$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

111.1.3a6

C

$3$

$0$

$0$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

111.1.3a7

C

$3$

$0$

$0$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

111.1.3a8

C

$3$

$0$

$0$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

111.1.3a9

C

$3$

$0$

$0$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

111.1.3a10

C

$3$

$0$

$0$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

111.1.3a11

C

$3$

$0$

$0$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

111.1.3a12

C

$3$

$0$

$0$

${\zeta }_{37}^{-7}+{\zeta }_{37}^3+{\zeta }_{37}^4$

${\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^7$

${\zeta }_{37}^{-14}+{\zeta }_{37}^6+{\zeta }_{37}^8$

${\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{14}$

${\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^2+{\zeta }_{37}^{15}$

${\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^5+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-11}+{\zeta }_{37}+{\zeta }_{37}^{10}$