Galois group: 38T2

• Label: 38T2
• Degree: $38$
• Order: $38$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{19}$

Group action invariants

Degree $n$:		$38$
Transitive number $t$:		$2$
Group:		$D_{19}$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$38$
Generators:		(1,4)(2,3)(5,37)(6,38)(7,36)(8,35)(9,33)(10,34)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,25)(18,26)(19,24)(20,23)(21,22), (1,9)(2,10)(3,8)(4,7)(5,6)(11,37)(12,38)(13,36)(14,35)(15,33)(16,34)(17,32)(18,31)(19,30)(20,29)(21,27)(22,28)(23,25)(24,26)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $D_{19}$

Low degree siblings

19T2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$1^{38}$	$1$	$1$	$()$
	$2^{19}$	$19$	$2$	$( 1, 2)( 3,37)( 4,38)( 5,36)( 6,35)( 7,34)( 8,33)( 9,31)(10,32)(11,30)(12,29) (13,27)(14,28)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$
	$19^{2}$	$2$	$19$	$( 1, 3, 5, 7,10,11,13,16,17,19,21,23,26,28,29,31,33,35,38)( 2, 4, 6, 8, 9,12, 14,15,18,20,22,24,25,27,30,32,34,36,37)$
	$19^{2}$	$2$	$19$	$( 1, 5,10,13,17,21,26,29,33,38, 3, 7,11,16,19,23,28,31,35)( 2, 6, 9,14,18,22, 25,30,34,37, 4, 8,12,15,20,24,27,32,36)$
	$19^{2}$	$2$	$19$	$( 1, 7,13,19,26,31,38, 5,11,17,23,29,35, 3,10,16,21,28,33)( 2, 8,14,20,25,32, 37, 6,12,18,24,30,36, 4, 9,15,22,27,34)$
	$19^{2}$	$2$	$19$	$( 1,10,17,26,33, 3,11,19,28,35, 5,13,21,29,38, 7,16,23,31)( 2, 9,18,25,34, 4, 12,20,27,36, 6,14,22,30,37, 8,15,24,32)$
	$19^{2}$	$2$	$19$	$( 1,11,21,31, 3,13,23,33, 5,16,26,35, 7,17,28,38,10,19,29)( 2,12,22,32, 4,14, 24,34, 6,15,25,36, 8,18,27,37, 9,20,30)$
	$19^{2}$	$2$	$19$	$( 1,13,26,38,11,23,35,10,21,33, 7,19,31, 5,17,29, 3,16,28)( 2,14,25,37,12,24, 36, 9,22,34, 8,20,32, 6,18,30, 4,15,27)$
	$19^{2}$	$2$	$19$	$( 1,16,29, 5,19,33,10,23,38,13,28, 3,17,31, 7,21,35,11,26)( 2,15,30, 6,20,34, 9,24,37,14,27, 4,18,32, 8,22,36,12,25)$
	$19^{2}$	$2$	$19$	$( 1,17,33,11,28, 5,21,38,16,31,10,26, 3,19,35,13,29, 7,23)( 2,18,34,12,27, 6, 22,37,15,32, 9,25, 4,20,36,14,30, 8,24)$
	$19^{2}$	$2$	$19$	$( 1,19,38,17,35,16,33,13,31,11,29,10,28, 7,26, 5,23, 3,21)( 2,20,37,18,36,15, 34,14,32,12,30, 9,27, 8,25, 6,24, 4,22)$

Group invariants

Order:		$38=2 \cdot 19$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		38.1
Character table:

1A

2A

19A1

19A2

19A3

19A4

19A5

19A6

19A7

19A8

19A9

Size

1

19

2

2

2

2

2

2

2

2

2

2 P

1A

1A

19A4

19A7

19A6

19A3

19A9

19A8

19A1

19A5

19A2

19 P

1A

2A

19A6

19A1

19A9

19A5

19A4

19A7

19A8

19A2

19A3

Type

38.1.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

38.1.1b

R

$1$

$-1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

38.1.2a1

R

$2$

$0$

${\zeta }_{19}^{-9}+{\zeta }_{19}^9$

${\zeta }_{19}^{-1}+{\zeta }_{19}$

${\zeta }_{19}^{-8}+{\zeta }_{19}^8$

${\zeta }_{19}^{-2}+{\zeta }_{19}^2$

${\zeta }_{19}^{-7}+{\zeta }_{19}^7$

${\zeta }_{19}^{-3}+{\zeta }_{19}^3$

${\zeta }_{19}^{-6}+{\zeta }_{19}^6$

${\zeta }_{19}^{-4}+{\zeta }_{19}^4$

${\zeta }_{19}^{-5}+{\zeta }_{19}^5$

38.1.2a2

R

$2$

$0$

${\zeta }_{19}^{-8}+{\zeta }_{19}^8$

${\zeta }_{19}^{-3}+{\zeta }_{19}^3$

${\zeta }_{19}^{-5}+{\zeta }_{19}^5$

${\zeta }_{19}^{-6}+{\zeta }_{19}^6$

${\zeta }_{19}^{-2}+{\zeta }_{19}^2$

${\zeta }_{19}^{-9}+{\zeta }_{19}^9$

${\zeta }_{19}^{-1}+{\zeta }_{19}$

${\zeta }_{19}^{-7}+{\zeta }_{19}^7$

${\zeta }_{19}^{-4}+{\zeta }_{19}^4$

38.1.2a3

R

$2$

$0$

${\zeta }_{19}^{-7}+{\zeta }_{19}^7$

${\zeta }_{19}^{-5}+{\zeta }_{19}^5$

${\zeta }_{19}^{-2}+{\zeta }_{19}^2$

${\zeta }_{19}^{-9}+{\zeta }_{19}^9$

${\zeta }_{19}^{-3}+{\zeta }_{19}^3$

${\zeta }_{19}^{-4}+{\zeta }_{19}^4$

${\zeta }_{19}^{-8}+{\zeta }_{19}^8$

${\zeta }_{19}^{-1}+{\zeta }_{19}$

${\zeta }_{19}^{-6}+{\zeta }_{19}^6$

38.1.2a4

R

$2$

$0$

${\zeta }_{19}^{-6}+{\zeta }_{19}^6$

${\zeta }_{19}^{-7}+{\zeta }_{19}^7$

${\zeta }_{19}^{-1}+{\zeta }_{19}$

${\zeta }_{19}^{-5}+{\zeta }_{19}^5$

${\zeta }_{19}^{-8}+{\zeta }_{19}^8$

${\zeta }_{19}^{-2}+{\zeta }_{19}^2$

${\zeta }_{19}^{-4}+{\zeta }_{19}^4$

${\zeta }_{19}^{-9}+{\zeta }_{19}^9$

${\zeta }_{19}^{-3}+{\zeta }_{19}^3$

38.1.2a5

R

$2$

$0$

${\zeta }_{19}^{-5}+{\zeta }_{19}^5$

${\zeta }_{19}^{-9}+{\zeta }_{19}^9$

${\zeta }_{19}^{-4}+{\zeta }_{19}^4$

${\zeta }_{19}^{-1}+{\zeta }_{19}$

${\zeta }_{19}^{-6}+{\zeta }_{19}^6$

${\zeta }_{19}^{-8}+{\zeta }_{19}^8$

${\zeta }_{19}^{-3}+{\zeta }_{19}^3$

${\zeta }_{19}^{-2}+{\zeta }_{19}^2$

${\zeta }_{19}^{-7}+{\zeta }_{19}^7$

38.1.2a6

R

$2$

$0$

${\zeta }_{19}^{-4}+{\zeta }_{19}^4$

${\zeta }_{19}^{-8}+{\zeta }_{19}^8$

${\zeta }_{19}^{-7}+{\zeta }_{19}^7$

${\zeta }_{19}^{-3}+{\zeta }_{19}^3$

${\zeta }_{19}^{-1}+{\zeta }_{19}$

${\zeta }_{19}^{-5}+{\zeta }_{19}^5$

${\zeta }_{19}^{-9}+{\zeta }_{19}^9$

${\zeta }_{19}^{-6}+{\zeta }_{19}^6$

${\zeta }_{19}^{-2}+{\zeta }_{19}^2$

38.1.2a7

R

$2$

$0$

${\zeta }_{19}^{-3}+{\zeta }_{19}^3$

${\zeta }_{19}^{-6}+{\zeta }_{19}^6$

${\zeta }_{19}^{-9}+{\zeta }_{19}^9$

${\zeta }_{19}^{-7}+{\zeta }_{19}^7$

${\zeta }_{19}^{-4}+{\zeta }_{19}^4$

${\zeta }_{19}^{-1}+{\zeta }_{19}$

${\zeta }_{19}^{-2}+{\zeta }_{19}^2$

${\zeta }_{19}^{-5}+{\zeta }_{19}^5$

${\zeta }_{19}^{-8}+{\zeta }_{19}^8$

38.1.2a8

R

$2$

$0$

${\zeta }_{19}^{-2}+{\zeta }_{19}^2$

${\zeta }_{19}^{-4}+{\zeta }_{19}^4$

${\zeta }_{19}^{-6}+{\zeta }_{19}^6$

${\zeta }_{19}^{-8}+{\zeta }_{19}^8$

${\zeta }_{19}^{-9}+{\zeta }_{19}^9$

${\zeta }_{19}^{-7}+{\zeta }_{19}^7$

${\zeta }_{19}^{-5}+{\zeta }_{19}^5$

${\zeta }_{19}^{-3}+{\zeta }_{19}^3$

${\zeta }_{19}^{-1}+{\zeta }_{19}$

38.1.2a9

R

$2$

$0$

${\zeta }_{19}^{-1}+{\zeta }_{19}$

${\zeta }_{19}^{-2}+{\zeta }_{19}^2$

${\zeta }_{19}^{-3}+{\zeta }_{19}^3$

${\zeta }_{19}^{-4}+{\zeta }_{19}^4$

${\zeta }_{19}^{-5}+{\zeta }_{19}^5$

${\zeta }_{19}^{-6}+{\zeta }_{19}^6$

${\zeta }_{19}^{-7}+{\zeta }_{19}^7$

${\zeta }_{19}^{-8}+{\zeta }_{19}^8$

${\zeta }_{19}^{-9}+{\zeta }_{19}^9$