Galois group: 21T2

• Label: 21T2
• Degree: $21$
• Order: $21$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_7:C_3$

Group action invariants

Degree $n$:		$21$
Transitive number $t$:		$2$
Group:		$C_7:C_3$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$21$
Generators:		(1,5,10)(2,6,11)(3,4,12)(7,16,13)(8,17,14)(9,18,15)(19,20,21), (1,4,7,11,14,18,19)(2,5,8,12,15,16,20)(3,6,9,10,13,17,21)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $C_7:C_3$

Low degree siblings

7T3

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

Group invariants

Order:		$21=3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		21.1
Character table:

		1A	3A1	3A-1	7A1	7A-1
	Size	1	7	7	3	3
	3 P	1A	3A-1	3A1	7A1	7A-1
	7 P	1A	1A	1A	7A-1	7A1
	Type
21.1.1a	R	$1$	$1$	$1$	$1$	$1$
21.1.1b1	C	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
21.1.1b2	C	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
21.1.3a1	C	$3$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
21.1.3a2	C	$3$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$