Galois group: 7T3

• Label: 7T3
• Degree: $7$
• Order: $21$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_7:C_3$

Group action invariants

Degree $n$:		$7$
Transitive number $t$:		$3$
Group:		$C_7:C_3$
CHM label:		$F_{21}(7) = 7:3$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,4)(3,6,5), (1,2,3,4,5,6,7)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$3$:	$C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

21T2

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	$1^{7}$	$1$	$1$	$0$	$()$
3A1	$3^{2},1$	$7$	$3$	$4$	$(2,3,5)(4,7,6)$
3A-1	$3^{2},1$	$7$	$3$	$4$	$(2,5,3)(4,6,7)$
7A1	$7$	$3$	$7$	$6$	$(1,2,3,4,5,6,7)$
7A-1	$7$	$3$	$7$	$6$	$(1,4,7,3,6,2,5)$

Malle's constant $a(G)$:     $1/4$

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	$3$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$