Galois group: 35T31

• Label: 35T31
• Degree: $35$
• Order: $5040$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $S_7$

Group invariants

Abstract group:		$S_7$
Order:		$5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$35$
Transitive number $t$:		$31$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33)$, $(2,5)(3,7)(9,10)(13,14)(16,21)(17,22)(19,26)(27,33)(28,31)(30,32)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Degree 7: None

Low degree siblings

7T7, 14T46, 21T38, 30T565, 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	Index	Representative
1A	$1^{35}$	$1$	$1$	$0$	$()$
2A	$2^{10},1^{15}$	$21$	$2$	$10$	$( 1,35)( 2,10)( 3,32)( 5, 9)( 6,12)( 7,30)( 8,34)(17,19)(22,26)(23,24)$
2B	$2^{14},1^{7}$	$105$	$2$	$14$	$( 1,20)( 2,34)( 3,23)( 4, 5)( 7,25)( 8,11)(10,27)(12,13)(14,29)(15,24)(16,32)(17,28)(18,26)(19,35)$
2C	$2^{16},1^{3}$	$105$	$2$	$16$	$( 2,17)( 3, 7)( 4,20)( 5,22)( 6,23)( 8,15)( 9,13)(10,14)(11,12)(16,19)(18,24)(21,26)(27,30)(28,31)(29,35)(32,33)$
3A	$3^{10},1^{5}$	$70$	$3$	$20$	$( 1,11,24)( 3,14,19)( 7,13,26)( 8,15,20)( 9,21,31)(10,16,28)(12,18,25)(17,27,32)(22,33,30)(23,29,35)$
3B	$3^{11},1^{2}$	$280$	$3$	$22$	$( 1,34,25)( 2,33,11)( 3,29,20)( 4, 7,35)( 5,30,18)( 6,31,15)( 8,23,28)( 9,16,26)(12,17,32)(13,19,21)(22,27,24)$
4A	$4^{7},2^{3},1$	$210$	$4$	$24$	$( 1,28,20,17)( 2, 4,34, 5)( 3,25,23, 7)( 8,27,11,10)( 9,33)(12,29,13,14)(15,32,24,16)(18,35,26,19)(21,30)(22,31)$
4B	$4^{7},2^{3},1$	$630$	$4$	$24$	$( 1,27,22,32)( 2,25)( 3,28)( 4, 6)( 5,18,34,12)( 7,16,23,10)( 8,13,21,35)( 9,26,15,29)(11,33,30,24)(14,31,19,20)$
5A	$5^{7}$	$504$	$5$	$28$	$( 1,24,30,34,17)( 2,32,22, 5,12)( 3,26, 9, 6,10)( 4,14,28,21,13)( 7, 8,19,35,23)(11,25,33,27,18)(15,20,31,29,16)$
6A	$6^{4},3^{2},2^{2},1$	$210$	$6$	$26$	$( 1,15,11,20,24, 8)( 2,34)( 3,35,14,23,19,29)( 4, 5)( 7,18,13,25,26,12)( 9,31,21)(10,17,16,27,28,32)(22,30,33)$
6B	$6^{3},3^{4},2,1^{3}$	$420$	$6$	$24$	$( 1,10,31,11,28, 9)( 2, 4, 5)( 3,12, 7,14,25,13)( 8,17,33,20,27,22)(15,32,30)(16,21,24)(18,26,19)(23,29)$
6C	$6^{5},3,2$	$840$	$6$	$28$	$( 1,25,34)( 2,12,33,17,11,32)( 3, 4,29, 7,20,35)( 5,24,30,22,18,27)( 6, 8,31,23,15,28)( 9,21,16,13,26,19)(10,14)$
7A	$7^{5}$	$720$	$7$	$30$	$( 1,23, 8,17,24, 7,20)( 2, 6, 4,18,21,33,14)( 3,34,12,16,26,31,11)( 5,25,15,22,29,10,19)( 9,27,32,30,13,28,35)$
10A	$10^{2},5^{3}$	$504$	$10$	$30$	$( 1, 8,24,19,30,35,34,23,17, 7)( 2, 9,32, 6,22,10, 5, 3,12,26)( 4,21,14,13,28)(11,27,25,18,33)(15,29,20,16,31)$
12A	$12^{2},6,4,1$	$420$	$12$	$30$	$( 1,27,15,28,11,32,20,10,24,17, 8,16)( 2, 5,34, 4)( 3,13,35,25,14,26,23,12,19, 7,29,18)( 9,30,31,33,21,22)$

Malle's constant $a(G)$:     $1/10$

Character table

		1A	2A	2B	2C	3A	3B	4A	4B	5A	6A	6B	6C	7A	10A	12A
	Size	1	21	105	105	70	280	210	630	504	210	420	840	720	504	420
	2 P	1A	1A	1A	1A	3A	3B	2B	2B	5A	3A	3A	3B	7A	5A	6A
	3 P	1A	2A	2B	2C	1A	1A	4A	4B	5A	2B	2A	2C	7A	10A	4A
	5 P	1A	2A	2B	2C	3A	3B	4A	4B	1A	6A	6B	6C	7A	2A	12A
	7 P	1A	2A	2B	2C	3A	3B	4A	4B	5A	6A	6B	6C	1A	10A	12A
	Type
5040.w.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
5040.w.1b	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
5040.w.6a	R	$6$	$4$	$2$	$0$	$3$	$0$	$2$	$0$	$1$	$-1$	$1$	$0$	$-1$	$-1$	$-1$
5040.w.6b	R	$6$	$-4$	$2$	$0$	$3$	$0$	$-2$	$0$	$1$	$-1$	$-1$	$0$	$-1$	$1$	$1$
5040.w.14a	R	$14$	$6$	$2$	$2$	$2$	$-1$	$0$	$0$	$-1$	$2$	$0$	$-1$	$0$	$1$	$0$
5040.w.14b	R	$14$	$-4$	$2$	$0$	$-1$	$2$	$2$	$0$	$-1$	$-1$	$-1$	$0$	$0$	$1$	$-1$
5040.w.14c	R	$14$	$4$	$2$	$0$	$-1$	$2$	$-2$	$0$	$-1$	$-1$	$1$	$0$	$0$	$-1$	$1$
5040.w.14d	R	$14$	$-6$	$2$	$-2$	$2$	$-1$	$0$	$0$	$-1$	$2$	$0$	$1$	$0$	$-1$	$0$
5040.w.15a	R	$15$	$-5$	$-1$	$3$	$3$	$0$	$-1$	$-1$	$0$	$-1$	$1$	$0$	$1$	$0$	$-1$
5040.w.15b	R	$15$	$5$	$-1$	$-3$	$3$	$0$	$1$	$-1$	$0$	$-1$	$-1$	$0$	$1$	$0$	$1$
5040.w.20a	R	$20$	$0$	$-4$	$0$	$2$	$2$	$0$	$0$	$0$	$2$	$0$	$0$	$-1$	$0$	$0$
5040.w.21a	R	$21$	$-1$	$1$	$3$	$-3$	$0$	$1$	$-1$	$1$	$1$	$-1$	$0$	$0$	$-1$	$1$
5040.w.21b	R	$21$	$1$	$1$	$-3$	$-3$	$0$	$-1$	$-1$	$1$	$1$	$1$	$0$	$0$	$1$	$-1$
5040.w.35a	R	$35$	$5$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$0$	$-1$	$-1$	$1$	$0$	$0$	$-1$
5040.w.35b	R	$35$	$-5$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$0$	$-1$	$1$	$-1$	$0$	$0$	$1$

Regular extensions

Data not computed