Galois group: 31T3

• Label: 31T3
• Degree: $31$
• Order: $93$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{31}:C_{3}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$3$:	$C_3$

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

Group invariants

Order:		$93=3 \cdot 31$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		93.1
Character table:

1A

3A1

3A-1

31A1

31A-1

31A2

31A-2

31A3

31A-3

31A4

31A-4

31A8

31A-8

Size

1

31

31

3

3

3

3

3

3

3

3

3

3

3 P

1A

3A-1

3A1

31A4

31A-3

31A1

31A-4

31A-1

31A2

31A8

31A-2

31A-8

31A3

31 P

1A

1A

1A

31A-1

31A-4

31A-8

31A1

31A8

31A3

31A-2

31A-3

31A2

31A4

Type

93.1.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

93.1.1b1

C

$1$

${\zeta }_3^{-1}$

${\zeta }_3$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

93.1.1b2

C

$1$

${\zeta }_3$

${\zeta }_3^{-1}$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

93.1.3a1

C

$3$

$0$

$0$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}$

${\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$

${\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^4+{\zeta }_{31}^7$

${\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}^6$

${\zeta }_{31}^{-6}+{\zeta }_{31}+{\zeta }_{31}^5$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^2+{\zeta }_{31}^{10}$

93.1.3a2

C

$3$

$0$

$0$

${\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}$

${\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^4+{\zeta }_{31}^7$

${\zeta }_{31}^{-6}+{\zeta }_{31}+{\zeta }_{31}^5$

${\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}^6$

${\zeta }_{31}^{-12}+{\zeta }_{31}^2+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{12}$

93.1.3a3

C

$3$

$0$

$0$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}$

${\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-6}+{\zeta }_{31}+{\zeta }_{31}^5$

${\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}^6$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}$

${\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^2+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^4+{\zeta }_{31}^7$

${\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{11}$

93.1.3a4

C

$3$

$0$

$0$

${\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}^6$

${\zeta }_{31}^{-6}+{\zeta }_{31}+{\zeta }_{31}^5$

${\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^2+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^4+{\zeta }_{31}^7$

93.1.3a5

C

$3$

$0$

$0$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^2+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^4+{\zeta }_{31}^7$

${\zeta }_{31}^{-6}+{\zeta }_{31}+{\zeta }_{31}^5$

${\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}^6$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}$

${\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$

${\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}$

93.1.3a6

C

$3$

$0$

$0$

${\zeta }_{31}^{-12}+{\zeta }_{31}^2+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^4+{\zeta }_{31}^7$

${\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}^6$

${\zeta }_{31}^{-6}+{\zeta }_{31}+{\zeta }_{31}^5$

${\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}$

${\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$

93.1.3a7

C

$3$

$0$

$0$

${\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^4+{\zeta }_{31}^7$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}$

${\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^2+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{12}$

${\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}$

${\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}^6$

${\zeta }_{31}^{-6}+{\zeta }_{31}+{\zeta }_{31}^5$

93.1.3a8

C

$3$

$0$

$0$

${\zeta }_{31}^{-11}+{\zeta }_{31}^4+{\zeta }_{31}^7$

${\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{11}$

${\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^2+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}$

${\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-6}+{\zeta }_{31}+{\zeta }_{31}^5$

${\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}^6$

93.1.3a9

C

$3$

$0$

$0$

${\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}^6$

${\zeta }_{31}^{-6}+{\zeta }_{31}+{\zeta }_{31}^5$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{12}$

${\zeta }_{31}^{-12}+{\zeta }_{31}^2+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}$

${\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{11}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^4+{\zeta }_{31}^7$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}$

${\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$

93.1.3a10

C

$3$

$0$

$0$

${\zeta }_{31}^{-6}+{\zeta }_{31}+{\zeta }_{31}^5$

${\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}^6$

${\zeta }_{31}^{-12}+{\zeta }_{31}^2+{\zeta }_{31}^{10}$

${\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{12}$

${\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$

${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}$

${\zeta }_{31}^{-11}+{\zeta }_{31}^4+{\zeta }_{31}^7$

${\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{11}$

${\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$

${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}$