Galois group: 42T26

• Label: 42T26
• Degree: $42$
• Order: $168$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_8:C_3$

Group action invariants

Degree $n$:		$42$
Transitive number $t$:		$26$
Group:		$F_8:C_3$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		(1,25,39,2,26,40)(3,28,42)(4,27,41)(5,29,38)(6,30,37)(7,11,10,8,12,9)(13,34,24,14,33,23)(15,36,19,16,35,20)(17,32,21)(18,31,22), (1,13,27,38,8,22,36)(2,14,28,37,7,21,35)(3,16,29,39,10,23,31)(4,15,30,40,9,24,32)(5,17,26,42,12,19,33)(6,18,25,41,11,20,34)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

Degree 7: $C_7:C_3$

Degree 14: 14T11

Degree 21: 21T2

Low degree siblings

8T36, 14T11, 24T283, 28T27

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

Group invariants

Order:		$168=2^{3} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		168.43
Character table:

		1A	2A	3A1	3A-1	6A1	6A-1	7A1	7A-1
	Size	1	7	28	28	28	28	24	24
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	7A1	7A-1
	3 P	1A	2A	1A	1A	2A	2A	7A-1	7A1
	7 P	1A	2A	3A1	3A-1	6A1	6A-1	1A	1A
	Type
168.43.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
168.43.1b1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
168.43.1b2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
168.43.3a1	C	$3$	$3$	$0$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
168.43.3a2	C	$3$	$3$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
168.43.7a	R	$7$	$-1$	$1$	$1$	$-1$	$-1$	$0$	$0$
168.43.7b1	C	$7$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	$0$
168.43.7b2	C	$7$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$0$