Galois group: 42T2

• Label: 42T2
• Degree: $42$
• Order: $42$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_7:C_6$

Group action invariants

Degree $n$:		$42$
Transitive number $t$:		$2$
Group:		$C_7:C_6$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$42$
Generators:		(1,18,24,2,17,23)(3,14,19,4,13,20)(5,16,21,6,15,22)(7,41,32,8,42,31)(9,37,33,10,38,34)(11,40,35,12,39,36)(25,30,27,26,29,28), (1,7,13,22,28,36,38,2,8,14,21,27,35,37)(3,10,15,23,30,32,39,4,9,16,24,29,31,40)(5,12,17,20,26,34,41,6,11,18,19,25,33,42)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$
$21$:	$C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 7: $C_7:C_3$

Degree 14: $(C_7:C_3) \times C_2$

Degree 21: 21T2

Low degree siblings

14T5

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$7$	$3$	$( 1, 3, 5)( 2, 4, 6)( 7,16,25)( 8,15,26)( 9,17,28)(10,18,27)(11,13,30) (12,14,29)(19,38,31)(20,37,32)(21,39,33)(22,40,34)(23,42,36)(24,41,35)$
	$ 6, 6, 6, 6, 6, 6, 6 $	$7$	$6$	$( 1, 4, 5, 2, 3, 6)( 7,15,25, 8,16,26)( 9,18,28,10,17,27)(11,14,30,12,13,29) (19,37,31,20,38,32)(21,40,33,22,39,34)(23,41,36,24,42,35)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$7$	$3$	$( 1, 5, 3)( 2, 6, 4)( 7,25,16)( 8,26,15)( 9,28,17)(10,27,18)(11,30,13) (12,29,14)(19,31,38)(20,32,37)(21,33,39)(22,34,40)(23,36,42)(24,35,41)$
	$ 6, 6, 6, 6, 6, 6, 6 $	$7$	$6$	$( 1, 6, 3, 2, 5, 4)( 7,26,16, 8,25,15)( 9,27,17,10,28,18)(11,29,13,12,30,14) (19,32,38,20,31,37)(21,34,39,22,33,40)(23,35,42,24,36,41)$
	$ 14, 14, 14 $	$3$	$14$	$( 1, 7,13,22,28,36,38, 2, 8,14,21,27,35,37)( 3,10,15,23,30,32,39, 4, 9,16,24, 29,31,40)( 5,12,17,20,26,34,41, 6,11,18,19,25,33,42)$
	$ 7, 7, 7, 7, 7, 7 $	$3$	$7$	$( 1, 8,13,21,28,35,38)( 2, 7,14,22,27,36,37)( 3, 9,15,24,30,31,39) ( 4,10,16,23,29,32,40)( 5,11,17,19,26,33,41)( 6,12,18,20,25,34,42)$
	$ 7, 7, 7, 7, 7, 7 $	$3$	$7$	$( 1,21,38,13,35, 8,28)( 2,22,37,14,36, 7,27)( 3,24,39,15,31, 9,30) ( 4,23,40,16,32,10,29)( 5,19,41,17,33,11,26)( 6,20,42,18,34,12,25)$
	$ 14, 14, 14 $	$3$	$14$	$( 1,22,38,14,35, 7,28, 2,21,37,13,36, 8,27)( 3,23,39,16,31,10,30, 4,24,40,15, 32, 9,29)( 5,20,41,18,33,12,26, 6,19,42,17,34,11,25)$

Group invariants

Order:		$42=2 \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		42.2
Character table:

		1A	2A	3A1	3A-1	6A1	6A-1	7A1	7A-1	14A1	14A-1
	Size	1	1	7	7	7	7	3	3	3	3
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	7A1	7A-1	7A1	7A-1
	3 P	1A	2A	1A	1A	2A	2A	7A-1	7A1	14A-1	14A1
	7 P	1A	2A	3A1	3A-1	6A1	6A-1	1A	1A	2A	2A
	Type
42.2.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
42.2.1b	R	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
42.2.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$
42.2.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$
42.2.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$-1$	$-1$
42.2.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$-1$	$-1$
42.2.3a1	C	$3$	$3$	$0$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
42.2.3a2	C	$3$	$3$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
42.2.3b1	C	$3$	$-3$	$0$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$
42.2.3b2	C	$3$	$-3$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	$-{\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2$