Galois group: 21T4

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

Group invariants

Abstract group:		$F_7$
Order:		$42=2 \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$21$
Transitive number $t$:		$4$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{21}$	$1$	$1$	$0$	$()$
2A	$2^{9},1^{3}$	$7$	$2$	$9$	$( 1,12)( 2,11)( 3,10)( 4, 8)( 5, 7)( 6, 9)(13,21)(14,20)(15,19)$
3A1	$3^{7}$	$7$	$3$	$14$	$( 1,21, 7)( 2,19, 9)( 3,20, 8)( 4,10,14)( 5,12,13)( 6,11,15)(16,18,17)$
3A-1	$3^{7}$	$7$	$3$	$14$	$( 1, 7,21)( 2, 9,19)( 3, 8,20)( 4,14,10)( 5,13,12)( 6,15,11)(16,17,18)$
6A1	$6^{3},3$	$7$	$6$	$17$	$( 1, 5,21,12, 7,13)( 2, 6,19,11, 9,15)( 3, 4,20,10, 8,14)(16,17,18)$
6A-1	$6^{3},3$	$7$	$6$	$17$	$( 1,13, 7,12,21, 5)( 2,15, 9,11,19, 6)( 3,14, 8,10,20, 4)(16,18,17)$
7A	$7^{3}$	$6$	$7$	$18$	$( 1,15, 4,18, 8,19,12)( 2,14, 5,16, 7,20,11)( 3,13, 6,17, 9,21,10)$

Malle's constant $a(G)$:     $1/9$

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	$6$	$0$	$0$	$0$	$0$	$0$	$-1$

Regular extensions

Data not computed