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 invariants

Abstract group:		$C_{37}:C_{6}$
Order:		$222=2 \cdot 3 \cdot 37$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$37$
Transitive number $t$:		$5$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{37}$	$1$	$1$	$0$	$()$
2A	$2^{18},1$	$37$	$2$	$18$	$( 2,37)( 3,36)( 4,35)( 5,34)( 6,33)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$
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	$3^{12},1$	$37$	$3$	$24$	$( 2,27,11)( 3,16,21)( 4, 5,31)( 6,20,14)( 7, 9,24)( 8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)$
6A1	$6^{6},1$	$37$	$6$	$30$	$( 2,12,11,37,27,28)( 3,23,21,36,16,18)( 4,34,31,35, 5, 8)( 6,19,14,33,20,25)( 7,30,24,32, 9,15)(10,26,17,29,13,22)$
6A-1	$6^{6},1$	$37$	$6$	$30$	$( 2,28,27,37,11,12)( 3,18,16,36,21,23)( 4, 8, 5,35,31,34)( 6,25,20,33,14,19)( 7,15, 9,32,24,30)(10,22,13,29,17,26)$
37A1	$37$	$6$	$37$	$36$	$( 1,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
37A2	$37$	$6$	$37$	$36$	$( 1,36,34,32,30,28,26,24,22,20,18,16,14,12,10, 8, 6, 4, 2,37,35,33,31,29,27,25,23,21,19,17,15,13,11, 9, 7, 5, 3)$
37A3	$37$	$6$	$37$	$36$	$( 1,35,32,29,26,23,20,17,14,11, 8, 5, 2,36,33,30,27,24,21,18,15,12, 9, 6, 3,37,34,31,28,25,22,19,16,13,10, 7, 4)$
37A5	$37$	$6$	$37$	$36$	$( 1,33,28,23,18,13, 8, 3,35,30,25,20,15,10, 5,37,32,27,22,17,12, 7, 2,34,29,24,19,14, 9, 4,36,31,26,21,16,11, 6)$
37A6	$37$	$6$	$37$	$36$	$( 1,32,26,20,14, 8, 2,33,27,21,15, 9, 3,34,28,22,16,10, 4,35,29,23,17,11, 5,36,30,24,18,12, 6,37,31,25,19,13, 7)$
37A9	$37$	$6$	$37$	$36$	$( 1,29,20,11, 2,30,21,12, 3,31,22,13, 4,32,23,14, 5,33,24,15, 6,34,25,16, 7,35,26,17, 8,36,27,18, 9,37,28,19,10)$

Malle's constant $a(G)$:     $1/18$

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	37A2	37A3	37A6	37A1	37A9	37A5
	3 P	1A	2A	1A	1A	2A	2A	37A3	37A6	37A9	37A2	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	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$
222.1.6a2	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$
222.1.6a3	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$
222.1.6a4	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$
222.1.6a5	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$
222.1.6a6	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-7}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^7$	${\zeta }_{37}^{-14}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{14}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^{13}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^{10}+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{15}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$

Regular extensions

Data not computed