Galois group: 31T2

• Label: 31T2
• Degree: $31$
• Order: $62$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $D_{31}$

Group action invariants

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

Low degree resolvents

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

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

Group invariants

Order:		$62=2 \cdot 31$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		62.1
Character table:

1A

2A

31A1

31A2

31A3

31A4

31A5

31A6

31A7

31A8

31A9

31A10

31A11

31A12

31A13

31A14

31A15

Size

1

31

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2 P

1A

1A

31A1

31A13

31A3

31A7

31A10

31A11

31A6

31A15

31A5

31A2

31A8

31A12

31A4

31A14

31A9

31 P

1A

2A

31A14

31A4

31A11

31A5

31A15

31A1

31A9

31A7

31A8

31A3

31A12

31A13

31A6

31A10

31A2

Type

62.1.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

62.1.1b

R

$1$

$-1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

62.1.2a1

R

$2$

$0$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

62.1.2a2

R

$2$

$0$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

62.1.2a3

R

$2$

$0$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

62.1.2a4

R

$2$

$0$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

62.1.2a5

R

$2$

$0$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

62.1.2a6

R

$2$

$0$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

62.1.2a7

R

$2$

$0$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

62.1.2a8

R

$2$

$0$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

62.1.2a9

R

$2$

$0$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

62.1.2a10

R

$2$

$0$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

62.1.2a11

R

$2$

$0$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

62.1.2a12

R

$2$

$0$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

62.1.2a13

R

$2$

$0$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

62.1.2a14

R

$2$

$0$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

62.1.2a15

R

$2$

$0$

${\zeta }_{31}^{-1}+{\zeta }_{31}$

${\zeta }_{31}^{-2}+{\zeta }_{31}^2$

${\zeta }_{31}^{-3}+{\zeta }_{31}^3$

${\zeta }_{31}^{-4}+{\zeta }_{31}^4$

${\zeta }_{31}^{-5}+{\zeta }_{31}^5$

${\zeta }_{31}^{-6}+{\zeta }_{31}^6$

${\zeta }_{31}^{-7}+{\zeta }_{31}^7$

${\zeta }_{31}^{-8}+{\zeta }_{31}^8$

${\zeta }_{31}^{-9}+{\zeta }_{31}^9$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-13}+{\zeta }_{31}^{13}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{15}$