Galois group: 37T5

• Label: 37T5
• Degree: $37$
• Order: $222$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{37}:C_{6}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$

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

Group invariants

Order:		$222=2 \cdot 3 \cdot 37$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		222.1
Character table:

		1A	2A	3A1	3A-1	6A1	6A-1	37A1	37A2	37A3	37A5	37A6	37A9
	Size	1	37	37	37	37	37	6	6	6	6	6	6
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	37A1	37A3	37A2	37A6	37A9	37A5
	3 P	1A	2A	1A	1A	2A	2A	37A2	37A6	37A3	37A9	37A5	37A1
	37 P	1A	2A	3A1	3A-1	6A1	6A-1	1A	1A	1A	1A	1A	1A
	Type
222.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
222.1.1b	R	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
222.1.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$
222.1.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$
222.1.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$
222.1.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$
222.1.6a1	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$
222.1.6a2	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$
222.1.6a3	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$
222.1.6a4	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$
222.1.6a5	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$
222.1.6a6	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$