Galois group: 21T33

• Label: 21T33
• Degree: $21$
• Order: $2520$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $A_7$

Group invariants

Abstract group:		$A_7$
Order:		$2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$21$
Transitive number $t$:		$33$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15)$, $(4,6,5)(9,11,10)(13,15,14)(16,18,17)(19,20,21)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 7: None

Low degree siblings

7T6, 15T47 x 2, 35T28, 42T294, 42T299

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	$2^{8},1^{5}$	$105$	$2$	$8$	$( 1, 2)( 4, 5)( 8,12)( 9,14)(10,13)(11,15)(16,17)(20,21)$
3A	$3^{5},1^{6}$	$70$	$3$	$10$	$( 1, 4, 2)( 7, 9,13)( 8,16,12)(10,19,14)(11,20,15)$
3B	$3^{7}$	$280$	$3$	$14$	$( 1,11, 9)( 2,21,16)( 3,15,19)( 4, 6,20)( 5,18,13)( 7,10, 8)(12,14,17)$
4A	$4^{4},2^{2},1$	$630$	$4$	$14$	$( 1, 4, 2, 5)( 3, 6)( 7,19)( 8,20,12,21)( 9,13,14,10)(11,16,15,17)$
5A	$5^{4},1$	$504$	$5$	$16$	$( 1, 9,10,11, 8)( 2,13,14,15,12)( 3, 4,19,21,18)( 5,20,17, 6,16)$
6A	$6^{2},3,2^{2},1^{2}$	$210$	$6$	$14$	$( 1,14, 4,10, 2,19)( 3,21)( 6,17)( 7,13, 9)( 8,15,16,11,12,20)$
7A1	$7^{3}$	$360$	$7$	$18$	$( 1,13,17, 6, 7,19,18)( 2,14,21,11, 9,16, 3)( 4,12, 5,15,10,20, 8)$
7A-1	$7^{3}$	$360$	$7$	$18$	$( 1,18,19, 7, 6,17,13)( 2, 3,16, 9,11,21,14)( 4, 8,20,10,15, 5,12)$

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	3A	3B	4A	5A	6A	7A1	7A-1
	Size	1	105	70	280	630	504	210	360	360
	2 P	1A	1A	3A	3B	2A	5A	3A	7A1	7A-1
	3 P	1A	2A	1A	1A	4A	5A	2A	7A-1	7A1
	5 P	1A	2A	3A	3B	4A	1A	6A	7A-1	7A1
	7 P	1A	2A	3A	3B	4A	5A	6A	1A	1A
	Type
2520.a.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
2520.a.6a	R	$6$	$2$	$3$	$0$	$0$	$1$	$-1$	$-1$	$-1$
2520.a.10a1	C	$10$	$-2$	$1$	$1$	$0$	$0$	$1$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
2520.a.10a2	C	$10$	$-2$	$1$	$1$	$0$	$0$	$1$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
2520.a.14a	R	$14$	$2$	$-1$	$2$	$0$	$-1$	$-1$	$0$	$0$
2520.a.14b	R	$14$	$2$	$2$	$-1$	$0$	$-1$	$2$	$0$	$0$
2520.a.15a	R	$15$	$-1$	$3$	$0$	$-1$	$0$	$-1$	$1$	$1$
2520.a.21a	R	$21$	$1$	$-3$	$0$	$-1$	$1$	$1$	$0$	$0$
2520.a.35a	R	$35$	$-1$	$-1$	$-1$	$1$	$0$	$-1$	$0$	$0$

Regular extensions

Data not computed