Galois group: 38T6

• Label: 38T6
• Degree: $38$
• Order: $228$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{38}:C_6$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$C_6$ x 3
$12$:	$C_6\times C_2$
$114$:	$C_{19}:C_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Low degree siblings

38T6

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

Group invariants

Order:		$228=2^{2} \cdot 3 \cdot 19$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		228.7
Character table:

		1A	2A	2B	2C	3A1	3A-1	6A1	6A-1	6B1	6B-1	6C1	6C-1	19A1	19A2	19A4	38A1	38A3	38A9
	Size	1	1	19	19	19	19	19	19	19	19	19	19	6	6	6	6	6	6
	2 P	1A	1A	1A	1A	3A-1	3A1	3A1	3A-1	3A-1	3A1	3A1	3A-1	19A1	19A2	19A4	19A4	19A1	19A2
	3 P	1A	2A	2B	2C	1A	1A	2A	2A	2B	2B	2C	2C	19A1	19A2	19A4	38A1	38A3	38A9
	19 P	1A	2A	2B	2C	3A1	3A-1	6C-1	6C1	6A-1	6A1	6B1	6B-1	1A	1A	1A	2A	2A	2A
	Type
228.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
228.7.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
228.7.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
228.7.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
228.7.1e1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$
228.7.1e2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$
228.7.1f1	C	$1$	$-1$	$-1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$1$	$-1$	$-1$	$-1$
228.7.1f2	C	$1$	$-1$	$-1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$1$	$-1$	$-1$	$-1$
228.7.1g1	C	$1$	$-1$	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$1$	$-1$	$-1$	$-1$
228.7.1g2	C	$1$	$-1$	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$1$	$-1$	$-1$	$-1$
228.7.1h1	C	$1$	$1$	$-1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$
228.7.1h2	C	$1$	$1$	$-1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$
228.7.6a1	R	$6$	$6$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$
228.7.6a2	R	$6$	$6$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$
228.7.6a3	R	$6$	$6$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$
228.7.6b1	R	$6$	$-6$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	$-{\zeta }_{19}^{-5}-{\zeta }_{19}^{-3}-{\zeta }_{19}^{-2}-{\zeta }_{19}^2-{\zeta }_{19}^3-{\zeta }_{19}^5$	$-{\zeta }_{19}^{-9}-{\zeta }_{19}^{-6}-{\zeta }_{19}^{-4}-{\zeta }_{19}^4-{\zeta }_{19}^6-{\zeta }_{19}^9$	$-{\zeta }_{19}^{-8}-{\zeta }_{19}^{-7}-{\zeta }_{19}^{-1}-{\zeta }_{19}-{\zeta }_{19}^7-{\zeta }_{19}^8$
228.7.6b2	R	$6$	$-6$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	$-{\zeta }_{19}^{-9}-{\zeta }_{19}^{-6}-{\zeta }_{19}^{-4}-{\zeta }_{19}^4-{\zeta }_{19}^6-{\zeta }_{19}^9$	$-{\zeta }_{19}^{-8}-{\zeta }_{19}^{-7}-{\zeta }_{19}^{-1}-{\zeta }_{19}-{\zeta }_{19}^7-{\zeta }_{19}^8$	$-{\zeta }_{19}^{-5}-{\zeta }_{19}^{-3}-{\zeta }_{19}^{-2}-{\zeta }_{19}^2-{\zeta }_{19}^3-{\zeta }_{19}^5$
228.7.6b3	R	$6$	$-6$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{19}^{-5}+{\zeta }_{19}^{-3}+{\zeta }_{19}^{-2}+{\zeta }_{19}^2+{\zeta }_{19}^3+{\zeta }_{19}^5$	${\zeta }_{19}^{-9}+{\zeta }_{19}^{-6}+{\zeta }_{19}^{-4}+{\zeta }_{19}^4+{\zeta }_{19}^6+{\zeta }_{19}^9$	${\zeta }_{19}^{-8}+{\zeta }_{19}^{-7}+{\zeta }_{19}^{-1}+{\zeta }_{19}+{\zeta }_{19}^7+{\zeta }_{19}^8$	$-{\zeta }_{19}^{-8}-{\zeta }_{19}^{-7}-{\zeta }_{19}^{-1}-{\zeta }_{19}-{\zeta }_{19}^7-{\zeta }_{19}^8$	$-{\zeta }_{19}^{-5}-{\zeta }_{19}^{-3}-{\zeta }_{19}^{-2}-{\zeta }_{19}^2-{\zeta }_{19}^3-{\zeta }_{19}^5$	$-{\zeta }_{19}^{-9}-{\zeta }_{19}^{-6}-{\zeta }_{19}^{-4}-{\zeta }_{19}^4-{\zeta }_{19}^6-{\zeta }_{19}^9$