Galois group: 19T4

• Label: 19T4
• Degree: $19$
• Order: $114$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{19}:C_{6}$

Group action invariants

Degree $n$:		$19$
Transitive number $t$:		$4$
Group:		$C_{19}:C_{6}$
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), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11)

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	$1^{19}$	$1$	$1$	$0$	$()$
2A	$2^{9},1$	$19$	$2$	$9$	$( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$
3A1	$3^{6},1$	$19$	$3$	$12$	$( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$
3A-1	$3^{6},1$	$19$	$3$	$12$	$( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)$
6A1	$6^{3},1$	$19$	$6$	$15$	$( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)$
6A-1	$6^{3},1$	$19$	$6$	$15$	$( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)$
19A1	$19$	$6$	$19$	$18$	$( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
19A2	$19$	$6$	$19$	$18$	$( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)$
19A4	$19$	$6$	$19$	$18$	$( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)$

Malle's constant $a(G)$:     $1/9$

Group invariants

Order:		$114=2 \cdot 3 \cdot 19$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		114.1
Character table:

		1A	2A	3A1	3A-1	6A1	6A-1	19A1	19A2	19A4
	Size	1	19	19	19	19	19	6	6	6
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	19A2	19A4	19A1
	3 P	1A	2A	1A	1A	2A	2A	19A2	19A4	19A1
	19 P	1A	2A	3A1	3A-1	6A1	6A-1	1A	1A	1A
	Type
114.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
114.1.1b	R	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
114.1.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$
114.1.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$
114.1.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$1$
114.1.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$1$
114.1.6a1	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$
114.1.6a2	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$
114.1.6a3	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$