Galois group: 37T4

• Label: 37T4
• Degree: $37$
• Order: $148$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{37}:C_{4}$

Group invariants

Abstract group:		$C_{37}:C_{4}$
Order:		$148=2^{2} \cdot 37$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

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	$2^{18},1$	$37$	$2$	$18$	$( 1,36)( 2,35)( 3,34)( 4,33)( 5,32)( 6,31)( 7,30)( 8,29)( 9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)$
4A1	$4^{9},1$	$37$	$4$	$27$	$( 1,31,36, 6)( 2,25,35,12)( 3,19,34,18)( 4,13,33,24)( 5, 7,32,30)( 8,26,29,11)( 9,20,28,17)(10,14,27,23)(15,21,22,16)$
4A-1	$4^{9},1$	$37$	$4$	$27$	$( 1, 6,36,31)( 2,12,35,25)( 3,18,34,19)( 4,24,33,13)( 5,30,32, 7)( 8,11,29,26)( 9,17,28,20)(10,23,27,14)(15,16,22,21)$
37A1	$37$	$4$	$37$	$36$	$( 1,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
37A2	$37$	$4$	$37$	$36$	$( 1,36,34,32,30,28,26,24,22,20,18,16,14,12,10, 8, 6, 4, 2,37,35,33,31,29,27,25,23,21,19,17,15,13,11, 9, 7, 5, 3)$
37A3	$37$	$4$	$37$	$36$	$( 1,35,32,29,26,23,20,17,14,11, 8, 5, 2,36,33,30,27,24,21,18,15,12, 9, 6, 3,37,34,31,28,25,22,19,16,13,10, 7, 4)$
37A4	$37$	$4$	$37$	$36$	$( 1,34,30,26,22,18,14,10, 6, 2,35,31,27,23,19,15,11, 7, 3,36,32,28,24,20,16,12, 8, 4,37,33,29,25,21,17,13, 9, 5)$
37A5	$37$	$4$	$37$	$36$	$( 1,33,28,23,18,13, 8, 3,35,30,25,20,15,10, 5,37,32,27,22,17,12, 7, 2,34,29,24,19,14, 9, 4,36,31,26,21,16,11, 6)$
37A8	$37$	$4$	$37$	$36$	$( 1,30,22,14, 6,35,27,19,11, 3,32,24,16, 8,37,29,21,13, 5,34,26,18,10, 2,31,23,15, 7,36,28,20,12, 4,33,25,17, 9)$
37A9	$37$	$4$	$37$	$36$	$( 1,29,20,11, 2,30,21,12, 3,31,22,13, 4,32,23,14, 5,33,24,15, 6,34,25,16, 7,35,26,17, 8,36,27,18, 9,37,28,19,10)$
37A10	$37$	$4$	$37$	$36$	$( 1,28,18, 8,35,25,15, 5,32,22,12, 2,29,19, 9,36,26,16, 6,33,23,13, 3,30,20,10,37,27,17, 7,34,24,14, 4,31,21,11)$
37A15	$37$	$4$	$37$	$36$	$( 1,23, 8,30,15,37,22, 7,29,14,36,21, 6,28,13,35,20, 5,27,12,34,19, 4,26,11,33,18, 3,25,10,32,17, 2,24, 9,31,16)$

Malle's constant $a(G)$:     $1/18$

Character table

1A

2A

4A1

4A-1

37A1

37A2

37A3

37A4

37A5

37A8

37A9

37A10

37A15

Size

1

37

37

37

4

4

4

4

4

4

4

4

4

2 P

1A

1A

2A

2A

37A2

37A4

37A1

37A8

37A10

37A15

37A3

37A9

37A5

37 P

1A

2A

4A-1

4A1

37A3

37A1

37A9

37A2

37A15

37A4

37A10

37A5

37A8

Type

148.3.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

148.3.1b

R

$1$

$1$

$-1$

$-1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

148.3.1c1

C

$1$

$-1$

$-i$

$i$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

148.3.1c2

C

$1$

$-1$

$i$

$-i$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

148.3.4a1

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

148.3.4a2

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

148.3.4a3

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

148.3.4a4

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

148.3.4a5

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

148.3.4a6

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

148.3.4a7

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

148.3.4a8

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

148.3.4a9

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

Regular extensions

Data not computed