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 invariants

Abstract group:		$F_8:C_3$
Order:		$168=2^{3} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$42$
Transitive number $t$:		$26$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{42}$	$1$	$1$	$0$	$()$
2A	$2^{12},1^{18}$	$7$	$2$	$12$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(15,16)(17,18)(19,20)(25,26)(27,28)(31,32)(37,38)$
3A1	$3^{14}$	$28$	$3$	$28$	$( 1, 3, 5)( 2, 4, 6)( 7,16,25)( 8,15,26)( 9,18,27)(10,17,28)(11,13,30)(12,14,29)(19,37,32)(20,38,31)(21,40,34)(22,39,33)(23,41,36)(24,42,35)$
3A-1	$3^{14}$	$28$	$3$	$28$	$( 1, 5, 3)( 2, 6, 4)( 7,25,16)( 8,26,15)( 9,27,18)(10,28,17)(11,30,13)(12,29,14)(19,32,37)(20,31,38)(21,34,40)(22,33,39)(23,36,41)(24,35,42)$
6A1	$6^{4},3^{6}$	$28$	$6$	$32$	$( 1, 6, 3, 2, 5, 4)( 7,26,16, 8,25,15)( 9,28,18,10,27,17)(11,30,13)(12,29,14)(19,31,37,20,32,38)(21,34,40)(22,33,39)(23,36,41)(24,35,42)$
6A-1	$6^{4},3^{6}$	$28$	$6$	$32$	$( 1, 4, 5, 2, 3, 6)( 7,15,25, 8,16,26)( 9,17,27,10,18,28)(11,13,30)(12,14,29)(19,38,32,20,37,31)(21,40,34)(22,39,33)(23,41,36)(24,42,35)$
7A1	$7^{6}$	$24$	$7$	$36$	$( 1,28, 7,36,14,38,22)( 2,27, 8,35,13,37,21)( 3,29,10,32,15,39,24)( 4,30, 9,31,16,40,23)( 5,26,11,33,18,41,20)( 6,25,12,34,17,42,19)$
7A-1	$7^{6}$	$24$	$7$	$36$	$( 1,22,38,14,36, 7,28)( 2,21,37,13,35, 8,27)( 3,24,39,15,32,10,29)( 4,23,40,16,31, 9,30)( 5,20,41,18,33,11,26)( 6,19,42,17,34,12,25)$

Malle's constant $a(G)$:     $1/12$

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	$7$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$0$

Regular extensions

Data not computed