Galois group: 21T38

• Label: 21T38
• Degree: $21$
• Order: $5040$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $S_7$

Group action invariants

Degree $n$:		$21$
Transitive number $t$:		$38$
Group:		$S_7$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15), (2,3)(4,5,6)(7,8)(9,10,11)(13,17,15,16,14,18)(19,21,20)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 7: None

Low degree siblings

7T7, 14T46, 30T565, 35T31, 42T411, 42T412, 42T413, 42T418

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$21$	$2$	$( 2, 6)( 7,11)(12,18)(13,20)(14,21)$
	$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $	$70$	$3$	$( 1,10, 8)( 2,14,12)( 3, 5,17)( 4,19,16)( 6,21,18)$
	$ 6, 3, 3, 3, 2, 2, 1, 1 $	$420$	$6$	$( 1, 8,10)( 2,18,14, 6,12,21)( 3,17, 5)( 4,16,19)( 7,11)(13,20)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $	$105$	$2$	$( 1,16)( 2,12)( 4, 8)( 5,17)( 6,18)( 7,13)(10,19)(11,20)$
	$ 4, 4, 4, 4, 2, 1, 1, 1 $	$210$	$4$	$( 1, 8,16, 4)( 2, 7,12,13)( 3, 9)( 5,10,17,19)( 6,11,18,20)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$105$	$2$	$( 1,16)( 2,12)( 4, 8)( 5,18)( 6,17)( 7,13)(10,20)(11,19)(14,15)$
	$ 3, 3, 3, 3, 3, 3, 3 $	$280$	$3$	$( 1,11, 6)( 2, 7,15)( 3, 9,21)( 4,10,18)( 5, 8,20)(12,13,14)(16,19,17)$
	$ 6, 6, 3, 2, 2, 1, 1 $	$210$	$6$	$( 1,14, 8, 2,10,12)( 3, 5,17)( 4,21,16, 6,19,18)( 9,15)(11,13)$
	$ 12, 4, 3, 2 $	$420$	$12$	$( 1,16,14, 6, 8,19, 2,18,10, 4,12,21)( 3,17, 5)( 7,20)( 9,13,15,11)$
	$ 5, 5, 5, 5, 1 $	$504$	$5$	$( 1, 3,16,19,10)( 2,12,13,14, 7)( 4,17, 9, 5, 8)( 6,18,20,21,11)$
	$ 10, 5, 5, 1 $	$504$	$10$	$( 1,19, 3,10,16)( 2,21,12,11,13, 6,14,18, 7,20)( 4, 5,17, 8, 9)$
	$ 6, 6, 6, 3 $	$840$	$6$	$( 1, 4, 2, 6, 5, 3)( 7,20,14,18,10,16)( 8, 9,13,15,21,17)(11,19,12)$
	$ 4, 4, 4, 4, 2, 2, 1 $	$630$	$4$	$( 1, 8,16, 4)( 2, 7,12,13)( 3, 9)( 5,11,17,20)( 6,10,18,19)(14,15)$
	$ 7, 7, 7 $	$720$	$7$	$( 1,12, 4,18,19,11,14)( 2, 3,16,20,21,10, 7)( 5, 8,13, 6,17, 9,15)$

Group invariants

Order:		$5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		5040.w
Character table:

	Size
	2 P
	3 P
	5 P
	7 P
	Type