Galois group: 42T22

• Label: 42T22
• Degree: $42$
• Order: $126$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{21}:C_6$

Group invariants

Abstract group:		$C_{21}:C_6$
Order:		$126=2 \cdot 3^{2} \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$S_3$, $C_6$
$18$:	$S_3\times C_3$
$42$:	$F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $S_3\times C_3$

Degree 7: $F_7$

Degree 14: $F_7$

Degree 21: None

Low degree siblings

21T10, 42T18

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	$2^{21}$	$21$	$2$	$21$	$( 1,12)( 2,10)( 3,11)( 4, 7)( 5, 9)( 6, 8)(13,40)(14,42)(15,41)(16,39)(17,37)(18,38)(19,35)(20,34)(21,36)(22,32)(23,33)(24,31)(25,30)(26,29)(27,28)$
3A	$3^{14}$	$2$	$3$	$28$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,24,23)(25,27,26)(28,30,29)(31,32,33)(34,35,36)(37,38,39)(40,42,41)$
3B1	$3^{14}$	$7$	$3$	$28$	$( 1,19, 7)( 2,20, 8)( 3,21, 9)( 4,12,35)( 5,11,36)( 6,10,34)(13,27,31)(14,25,33)(15,26,32)(16,17,18)(22,41,29)(23,42,30)(24,40,28)(37,38,39)$
3B-1	$3^{14}$	$7$	$3$	$28$	$( 1, 7,19)( 2, 8,20)( 3, 9,21)( 4,35,12)( 5,36,11)( 6,34,10)(13,31,27)(14,33,25)(15,32,26)(16,18,17)(22,29,41)(23,30,42)(24,28,40)(37,39,38)$
3C1	$3^{13},1^{3}$	$14$	$3$	$26$	$( 1,31, 8)( 2,32, 9)( 3,33, 7)( 4,18,40)( 5,17,42)( 6,16,41)(10,28,24)(11,29,22)(12,30,23)(19,25,39)(20,27,37)(21,26,38)(34,36,35)$
3C-1	$3^{13},1^{3}$	$14$	$3$	$26$	$( 1, 8,31)( 2, 9,32)( 3, 7,33)( 4,40,18)( 5,42,17)( 6,41,16)(10,24,28)(11,22,29)(12,23,30)(19,39,25)(20,37,27)(21,38,26)(34,35,36)$
6A1	$6^{7}$	$21$	$6$	$35$	$( 1, 4,19,12, 7,35)( 2, 6,20,10, 8,34)( 3, 5,21,11, 9,36)(13,24,27,40,31,28)(14,23,25,42,33,30)(15,22,26,41,32,29)(16,38,17,39,18,37)$
6A-1	$6^{7}$	$21$	$6$	$35$	$( 1,35, 7,12,19, 4)( 2,34, 8,10,20, 6)( 3,36, 9,11,21, 5)(13,28,31,40,27,24)(14,30,33,42,25,23)(15,29,32,41,26,22)(16,37,18,39,17,38)$
7A	$7^{6}$	$6$	$7$	$36$	$( 1,31,21, 8,38,26,14)( 2,32,19, 9,39,25,13)( 3,33,20, 7,37,27,15)( 4,34,23,11,41,28,17)( 5,35,22,10,40,30,16)( 6,36,24,12,42,29,18)$
21A1	$21^{2}$	$6$	$21$	$40$	$( 1,25, 7,31,13,37,21, 2,27, 8,32,15,38,19, 3,26, 9,33,14,39,20)( 4,30,12,34,16,42,23, 5,29,11,35,18,41,22, 6,28,10,36,17,40,24)$
21A2	$21^{2}$	$6$	$21$	$40$	$( 1, 7,13,21,27,32,38, 3, 9,14,20,25,31,37, 2, 8,15,19,26,33,39)( 4,12,16,23,29,35,41, 6,10,17,24,30,34,42, 5,11,18,22,28,36,40)$

Malle's constant $a(G)$:     $1/21$

Character table

		1A	2A	3A	3B1	3B-1	3C1	3C-1	6A1	6A-1	7A	21A1	21A2
	Size	1	21	2	7	7	14	14	21	21	6	6	6
	2 P	1A	1A	3A	3B-1	3B1	3C-1	3C1	3B1	3B-1	7A	21A2	21A1
	3 P	1A	2A	1A	1A	1A	1A	1A	2A	2A	7A	7A	7A
	7 P	1A	2A	3A	3B1	3B-1	3C1	3C-1	6A1	6A-1	1A	3A	3A
	Type
126.9.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
126.9.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
126.9.1c1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$
126.9.1c2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$
126.9.1d1	C	$1$	$-1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$1$
126.9.1d2	C	$1$	$-1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$1$
126.9.2a	R	$2$	$0$	$-1$	$2$	$2$	$-1$	$-1$	$0$	$0$	$2$	$-1$	$-1$
126.9.2b1	C	$2$	$0$	$-1$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	$0$	$2$	$-1$	$-1$
126.9.2b2	C	$2$	$0$	$-1$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$0$	$2$	$-1$	$-1$
126.9.6a	R	$6$	$0$	$6$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$
126.9.6b1	R	$6$	$0$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	${\zeta }_{21}^{-10}-{\zeta }_{21}+2{\zeta }_{21}^2-{\zeta }_{21}^4-{\zeta }_{21}^7+{\zeta }_{21}^8+{\zeta }_{21}^9$	$-{\zeta }_{21}^{-10}+1+{\zeta }_{21}-2{\zeta }_{21}^2+{\zeta }_{21}^4+{\zeta }_{21}^7-{\zeta }_{21}^8-{\zeta }_{21}^9$
126.9.6b2	R	$6$	$0$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-{\zeta }_{21}^{-10}+1+{\zeta }_{21}-2{\zeta }_{21}^2+{\zeta }_{21}^4+{\zeta }_{21}^7-{\zeta }_{21}^8-{\zeta }_{21}^9$	${\zeta }_{21}^{-10}-{\zeta }_{21}+2{\zeta }_{21}^2-{\zeta }_{21}^4-{\zeta }_{21}^7+{\zeta }_{21}^8+{\zeta }_{21}^9$

Regular extensions

Data not computed