Galois group: 21T11

• Label: 21T11
• Degree: $21$
• Order: $126$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{21}:C_6$

Group action invariants

Degree $n$:		$21$
Transitive number $t$:		$11$
Group:		$C_{21}:C_6$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,12,5)(2,10,6)(3,11,4)(7,15,17)(8,13,18)(9,14,16)(19,21,20), (1,6,16,3,4,18)(2,5,17)(7,9)(10,21,13,12,19,15)(11,20,14)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$S_3$, $C_6$
$18$:	$S_3\times C_3$
$21$:	$C_7:C_3$
$42$:	$(C_7:C_3) \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $C_7:C_3$

Low degree siblings

42T19, 42T23

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 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $	$7$	$3$	$( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,19,16)(11,20,17)(12,21,18)$
	$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $	$7$	$3$	$( 4,13, 7)( 5,14, 8)( 6,15, 9)(10,16,19)(11,17,20)(12,18,21)$
	$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $	$3$	$2$	$( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)$
	$ 6, 6, 3, 3, 2, 1 $	$21$	$6$	$( 2, 3)( 4, 7,13)( 5, 9,14, 6, 8,15)(10,19,16)(11,21,17,12,20,18)$
	$ 6, 6, 3, 3, 2, 1 $	$21$	$6$	$( 2, 3)( 4,13, 7)( 5,15, 8, 6,14, 9)(10,16,19)(11,18,20,12,17,21)$
	$ 3, 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
	$ 3, 3, 3, 3, 3, 3, 3 $	$14$	$3$	$( 1, 2, 3)( 4, 8,15)( 5, 9,13)( 6, 7,14)(10,20,18)(11,21,16)(12,19,17)$
	$ 3, 3, 3, 3, 3, 3, 3 $	$14$	$3$	$( 1, 2, 3)( 4,14, 9)( 5,15, 7)( 6,13, 8)(10,17,21)(11,18,19)(12,16,20)$
	$ 7, 7, 7 $	$3$	$7$	$( 1, 4, 7,10,13,16,19)( 2, 5, 8,11,14,17,20)( 3, 6, 9,12,15,18,21)$
	$ 14, 7 $	$9$	$14$	$( 1, 4, 7,10,13,16,19)( 2, 6, 8,12,14,18,20, 3, 5, 9,11,15,17,21)$
	$ 21 $	$6$	$21$	$( 1, 5, 9,10,14,18,19, 2, 6, 7,11,15,16,20, 3, 4, 8,12,13,17,21)$
	$ 7, 7, 7 $	$3$	$7$	$( 1,10,19, 7,16, 4,13)( 2,11,20, 8,17, 5,14)( 3,12,21, 9,18, 6,15)$
	$ 14, 7 $	$9$	$14$	$( 1,10,19, 7,16, 4,13)( 2,12,20, 9,17, 6,14, 3,11,21, 8,18, 5,15)$
	$ 21 $	$6$	$21$	$( 1,11,21, 7,17, 6,13, 2,12,19, 8,18, 4,14, 3,10,20, 9,16, 5,15)$

Group invariants

Order:		$126=2 \cdot 3^{2} \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		126.8
Character table:

		1A	2A	3A	3B1	3B-1	3C1	3C-1	6A1	6A-1	7A1	7A-1	14A1	14A-1	21A1	21A-1
	Size	1	3	2	7	7	14	14	21	21	3	3	9	9	6	6
	2 P	1A	1A	3A	3B-1	3B1	3C-1	3C1	3B1	3B-1	7A1	7A-1	7A-1	7A1	21A1	21A-1
	3 P	1A	2A	1A	1A	1A	1A	1A	2A	2A	7A-1	7A1	14A-1	14A1	7A1	7A-1
	7 P	1A	2A	3A	3B1	3B-1	3C1	3C-1	6A1	6A-1	1A	1A	2A	2A	3A	3A
	Type
126.8.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
126.8.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$
126.8.1c1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$
126.8.1c2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$
126.8.1d1	C	$1$	$-1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$-1$	$-1$	$1$	$1$
126.8.1d2	C	$1$	$-1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$-1$	$-1$	$1$	$1$
126.8.2a	R	$2$	$0$	$-1$	$2$	$2$	$-1$	$-1$	$0$	$0$	$2$	$2$	$0$	$0$	$-1$	$-1$
126.8.2b1	C	$2$	$0$	$-1$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	$0$	$2$	$2$	$0$	$0$	$-1$	$-1$
126.8.2b2	C	$2$	$0$	$-1$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$0$	$2$	$2$	$0$	$0$	$-1$	$-1$
126.8.3a1	C	$3$	$3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
126.8.3a2	C	$3$	$3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
126.8.3b1	C	$3$	$-3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
126.8.3b2	C	$3$	$-3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
126.8.6a1	C	$6$	$0$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$	$-2{\zeta }_7^{-3}-2-2{\zeta }_7-2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2{\zeta }_7+2{\zeta }_7^2$	$0$	$0$	$-{\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2$
126.8.6a2	C	$6$	$0$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$	$2{\zeta }_7^{-3}+2{\zeta }_7+2{\zeta }_7^2$	$-2{\zeta }_7^{-3}-2-2{\zeta }_7-2{\zeta }_7^2$	$0$	$0$	${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$