Galois group: 37T7

• Label: 37T7
• Degree: $37$
• Order: $444$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{37}:C_{12}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$4$:	$C_4$
$6$:	$C_6$
$12$:	$C_{12}$

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

Group invariants

Order:		$444=2^{2} \cdot 3 \cdot 37$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		444.7
Character table:

		1A	2A	3A1	3A-1	4A1	4A-1	6A1	6A-1	12A1	12A-1	12A5	12A-5	37A1	37A2	37A3
	Size	1	37	37	37	37	37	37	37	37	37	37	37	12	12	12
	2 P	1A	1A	3A-1	3A1	2A	2A	3A1	3A-1	6A1	6A-1	6A-1	6A1	37A3	37A1	37A2
	3 P	1A	2A	1A	1A	4A-1	4A1	2A	2A	4A1	4A-1	4A1	4A-1	37A1	37A2	37A3
	37 P	1A	2A	3A1	3A-1	4A1	4A-1	6A1	6A-1	12A1	12A-1	12A5	12A-5	1A	1A	1A
	Type
444.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
444.7.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
444.7.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$
444.7.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$
444.7.1d1	C	$1$	$-1$	$1$	$1$	$-i$	$i$	$-1$	$-1$	$i$	$-i$	$i$	$-i$	$1$	$1$	$1$
444.7.1d2	C	$1$	$-1$	$1$	$1$	$i$	$-i$	$-1$	$-1$	$-i$	$i$	$-i$	$i$	$1$	$1$	$1$
444.7.1e1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$1$
444.7.1e2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$1$
444.7.1f1	C	$1$	$-1$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	$-{\zeta }_{12}^3$	${\zeta }_{12}^3$	$-{\zeta }_{12}^4$	${\zeta }_{12}^2$	$-{\zeta }_{12}^5$	${\zeta }_{12}$	$-{\zeta }_{12}$	${\zeta }_{12}^5$	$1$	$1$	$1$
444.7.1f2	C	$1$	$-1$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	${\zeta }_{12}^3$	$-{\zeta }_{12}^3$	${\zeta }_{12}^2$	$-{\zeta }_{12}^4$	${\zeta }_{12}$	$-{\zeta }_{12}^5$	${\zeta }_{12}^5$	$-{\zeta }_{12}$	$1$	$1$	$1$
444.7.1f3	C	$1$	$-1$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	${\zeta }_{12}^3$	$-{\zeta }_{12}^3$	$-{\zeta }_{12}^4$	${\zeta }_{12}^2$	${\zeta }_{12}^5$	$-{\zeta }_{12}$	${\zeta }_{12}$	$-{\zeta }_{12}^5$	$1$	$1$	$1$
444.7.1f4	C	$1$	$-1$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	$-{\zeta }_{12}^3$	${\zeta }_{12}^3$	${\zeta }_{12}^2$	$-{\zeta }_{12}^4$	$-{\zeta }_{12}$	${\zeta }_{12}^5$	$-{\zeta }_{12}^5$	${\zeta }_{12}$	$1$	$1$	$1$
444.7.12a1	R	$12$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^5+{\zeta }_{37}^7+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}+{\zeta }_{37}^{14}$
444.7.12a2	R	$12$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^5+{\zeta }_{37}^7+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$
444.7.12a3	R	$12$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^5+{\zeta }_{37}^7+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}+{\zeta }_{37}^{17}$