Galois group: 21T8

• Label: 21T8
• Degree: $21$
• Order: $84$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $S_3\times D_7$

Group action invariants

Degree $n$:		$21$
Transitive number $t$:		$8$
Group:		$S_3\times D_7$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3)(4,20,6,19,5,21)(7,17,9,16,8,18)(10,14,12,13,11,15), (1,17,10,5,19,14,7,2,16,11,4,20,13,8)(3,18,12,6,21,15,9)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$
$12$:	$D_{6}$
$14$:	$D_{7}$
$28$:	$D_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $D_{7}$

Low degree siblings

42T13, 42T14, 42T15

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

Group invariants

Order:		$84=2^{2} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		84.8
Character table:

		1A	2A	2B	2C	3A	6A	7A1	7A2	7A3	14A1	14A3	14A5	21A1	21A2	21A4
	Size	1	3	7	21	2	14	2	2	2	6	6	6	4	4	4
	2 P	1A	1A	1A	1A	3A	3A	7A3	7A1	7A2	7A1	7A3	7A2	21A2	21A4	21A1
	3 P	1A	2A	2B	2C	1A	2B	7A1	7A2	7A3	14A3	14A5	14A1	7A1	7A2	7A3
	7 P	1A	2A	2B	2C	3A	6A	1A	1A	1A	2A	2A	2A	3A	3A	3A
	Type
84.8.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
84.8.1b	R	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
84.8.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
84.8.1d	R	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
84.8.2a	R	$2$	$0$	$2$	$0$	$-1$	$-1$	$2$	$2$	$2$	$0$	$0$	$0$	$-1$	$-1$	$-1$
84.8.2b	R	$2$	$0$	$-2$	$0$	$-1$	$1$	$2$	$2$	$2$	$0$	$0$	$0$	$-1$	$-1$	$-1$
84.8.2c1	R	$2$	$2$	$0$	$0$	$2$	$0$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$
84.8.2c2	R	$2$	$2$	$0$	$0$	$2$	$0$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$
84.8.2c3	R	$2$	$2$	$0$	$0$	$2$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$
84.8.2d1	R	$2$	$-2$	$0$	$0$	$2$	$0$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$
84.8.2d2	R	$2$	$-2$	$0$	$0$	$2$	$0$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$
84.8.2d3	R	$2$	$-2$	$0$	$0$	$2$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$
84.8.4a1	R	$4$	$0$	$0$	$0$	$-2$	$0$	$2{\zeta }_7^{-3}+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2{\zeta }_7$	$2{\zeta }_7^{-2}+2{\zeta }_7^2$	$0$	$0$	$0$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-3}-{\zeta }_7^3$
84.8.4a2	R	$4$	$0$	$0$	$0$	$-2$	$0$	$2{\zeta }_7^{-2}+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2{\zeta }_7$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-2}-{\zeta }_7^2$
84.8.4a3	R	$4$	$0$	$0$	$0$	$-2$	$0$	$2{\zeta }_7^{-1}+2{\zeta }_7$	$2{\zeta }_7^{-2}+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2{\zeta }_7^3$	$0$	$0$	$0$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-1}-{\zeta }_7$