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 invariants

Abstract group:		$C_{19}:C_6$
Order:		$114=2 \cdot 3 \cdot 19$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$38$
Transitive number $t$:		$5$
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

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$

Resolvents shown for degrees $\leq 47$

Subfields

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	Index	Representative
1A	$1^{38}$	$1$	$1$	$0$	$()$
2A	$2^{19}$	$19$	$2$	$19$	$( 1,24)( 2,23)( 3,22)( 4,21)( 5,20)( 6,19)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)(25,37)(26,38)(27,36)(28,35)(29,33)(30,34)(31,32)$
3A1	$3^{12},1^{2}$	$19$	$3$	$24$	$( 1, 5,11)( 2, 6,12)( 3,28,26)( 4,27,25)( 7,34,15)( 8,33,16)( 9,17,30)(10,18,29)(13,24,20)(14,23,19)(21,36,37)(22,35,38)$
3A-1	$3^{12},1^{2}$	$19$	$3$	$24$	$( 1,11, 5)( 2,12, 6)( 3,26,28)( 4,25,27)( 7,15,34)( 8,16,33)( 9,30,17)(10,29,18)(13,20,24)(14,19,23)(21,37,36)(22,38,35)$
6A1	$6^{6},2$	$19$	$6$	$31$	$( 1,13, 5,24,11,20)( 2,14, 6,23,12,19)( 3,38,28,22,26,35)( 4,37,27,21,25,36)( 7, 9,34,17,15,30)( 8,10,33,18,16,29)(31,32)$
6A-1	$6^{6},2$	$19$	$6$	$31$	$( 1,20,11,24, 5,13)( 2,19,12,23, 6,14)( 3,35,26,22,28,38)( 4,36,25,21,27,37)( 7,30,15,17,34, 9)( 8,29,16,18,33,10)(31,32)$
19A1	$19^{2}$	$6$	$19$	$36$	$( 1,27,15, 4,29,18, 5,32,19, 7,34,22,10,35,23,11,38,25,14)( 2,28,16, 3,30,17, 6,31,20, 8,33,21, 9,36,24,12,37,26,13)$
19A2	$19^{2}$	$6$	$19$	$36$	$( 1,15,29, 5,19,34,10,23,38,14,27, 4,18,32, 7,22,35,11,25)( 2,16,30, 6,20,33, 9,24,37,13,28, 3,17,31, 8,21,36,12,26)$
19A4	$19^{2}$	$6$	$19$	$36$	$( 1,29,19,10,38,27,18, 7,35,25,15, 5,34,23,14, 4,32,22,11)( 2,30,20, 9,37,28,17, 8,36,26,16, 6,33,24,13, 3,31,21,12)$

Malle's constant $a(G)$:     $1/19$

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

Regular extensions

Data not computed