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 invariants

Abstract group:		$C_{37}:C_{9}$
Order:		$333=3^{2} \cdot 37$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

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

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

Regular extensions

Data not computed