Galois group: 21T4

• Label: 21T4
• Degree: $21$
• Order: $42$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_7$

Group action invariants

Degree $n$:		$21$
Transitive number $t$:		$4$
Group:		$F_7$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		(1,9,5,18,10,14)(2,8,6,16,12,13)(3,7,4,17,11,15)(19,21,20), (1,4,8,12,15,18,19)(2,5,7,11,14,16,20)(3,6,9,10,13,17,21)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $F_7$

Low degree siblings

7T4, 14T4, 42T4

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

Group invariants

Order:		$42=2 \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		42.1
Character table:

		1A	2A	3A1	3A-1	6A1	6A-1	7A
	Size	1	7	7	7	7	7	6
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	7A
	3 P	1A	2A	1A	1A	2A	2A	7A
	7 P	1A	2A	3A1	3A-1	6A1	6A-1	1A
	Type
42.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
42.1.1b	R	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
42.1.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$
42.1.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$
42.1.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$
42.1.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$
42.1.6a	R	$6$	$0$	$0$	$0$	$0$	$0$	$-1$