Galois group: 38T5

• Label: 38T5
• Degree: $38$
• Order: $114$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{19}:C_6$

Group action invariants

Degree $n$:		$38$
Transitive number $t$:		$5$
Group:		$C_{19}:C_6$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,34,29)(2,33,30)(3,9,13)(4,10,14)(5,23,35)(6,24,36)(7,38,19)(8,37,20)(11,27,25)(12,28,26)(15,18,32)(16,17,31), (1,3,27,12,10,24)(2,4,28,11,9,23)(5,13,34,8,38,17)(6,14,33,7,37,18)(15,20,29,36,32,21)(16,19,30,35,31,22)(25,26)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $C_{19}:C_{6}$

Low degree siblings

19T4

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

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	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$