Galois group: 35T36

• Label: 35T36
• Degree: $35$
• Order: $20160$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $A_8$

Group invariants

Abstract group:		$A_8$
Order:		$20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$35$
Transitive number $t$:		$36$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33)$, $(1,3,7)(2,6,5)(9,10,12)(11,14,13)(16,18,21)(17,22,23)(19,26,24)(25,31,28)(27,33,29)(30,35,32)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Degree 7: None

Low degree siblings

8T49, 15T72 x 2, 28T433

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	$2^{12},1^{11}$	$105$	$2$	$12$	$( 1,13)( 2,27)( 3,16)( 4,32)( 6,17)( 8,30)( 9,24)(10,25)(14,18)(20,35)(22,29)(26,31)$
2B	$2^{14},1^{7}$	$210$	$2$	$14$	$( 1,17)( 2,23)( 3,34)( 4,25)( 5, 7)( 8,32)( 9,30)(10,24)(11,16)(14,29)(15,27)(19,35)(20,28)(21,33)$
3A	$3^{10},1^{5}$	$112$	$3$	$20$	$( 1,18,17)( 2,33,34)( 3,21,23)( 4,10, 8)( 5,11,27)( 6,14,29)( 7,15,16)(19,31,35)(20,28,26)(24,25,32)$
3B	$3^{11},1^{2}$	$1120$	$3$	$22$	$( 1,30,26)( 2,32,18)( 3,25, 6)( 4,14,27)( 5,15,33)( 8,31,13)( 9,20,29)(10,17,16)(11,34,21)(12,19,28)(22,24,35)$
4A	$4^{6},2^{4},1^{3}$	$1260$	$4$	$22$	$( 1, 7,25,28)( 2,15,23,27)( 4,35,17,33)( 5,30,20,29)( 6,32,22, 8)( 9,19,14,21)(10,18)(11,16)(12,26)(13,24)$
4B	$4^{7},2^{3},1$	$2520$	$4$	$24$	$( 1, 5,17, 7)( 2,27,23,15)( 3,11,34,16)( 4,19,25,35)( 6,18)( 8,24,32,10)( 9,20,30,28)(12,26)(13,22)(14,33,29,21)$
5A	$5^{7}$	$1344$	$5$	$28$	$( 1,28,16,19, 8)( 2,27,24, 3,29)( 4,18,26, 7,31)( 5,25,21, 6,33)( 9,12,22,13,30)(10,17,20,15,35)(11,32,23,14,34)$
6A	$6^{4},3^{2},2^{2},1$	$1680$	$6$	$26$	$( 1,17, 5,27,11,18)( 2,19)( 3,14, 6)( 4, 8,10,26,28,20)( 7,32,13,25,15,12)( 9,33,35,22,31,34)(16,24)(21,29,23)$
6B	$6^{4},3^{3},1^{2}$	$3360$	$6$	$26$	$( 1,31,30,13,26, 8)( 2,14,32,27,18, 4)( 3,17,25,16, 6,10)( 5,33,15)( 9,22,20,24,29,35)(11,21,34)(12,28,19)$
7A1	$7^{5}$	$2880$	$7$	$30$	$( 1,20,27, 6,21,10,30)( 2,31,11,19, 5,14, 9)( 3, 4,33,35,24,17,13)( 7,15, 8,34,29,23,16)(12,22,28,25,18,26,32)$
7A-1	$7^{5}$	$2880$	$7$	$30$	$( 1,30,10,21, 6,27,20)( 2, 9,14, 5,19,11,31)( 3,13,17,24,35,33, 4)( 7,16,23,29,34, 8,15)(12,32,26,18,25,28,22)$
15A1	$15^{2},5$	$1344$	$15$	$32$	$( 1,15, 4,28,35,18,16,10,26,19,17, 7, 8,20,31)( 2,32, 6,27,23,33,24,14, 5, 3,34,25,29,11,21)( 9,22,30,12,13)$
15A-1	$15^{2},5$	$1344$	$15$	$32$	$( 1,31,20, 8, 7,17,19,26,10,16,18,35,28, 4,15)( 2,21,11,29,25,34, 3, 5,14,24,33,23,27, 6,32)( 9,13,12,30,22)$

Malle's constant $a(G)$:     $1/12$

Character table

		1A	2A	2B	3A	3B	4A	4B	5A	6A	6B	7A1	7A-1	15A1	15A-1
	Size	1	105	210	112	1120	1260	2520	1344	1680	3360	2880	2880	1344	1344
	2 P	1A	1A	1A	3A	3B	2A	2B	5A	3A	3B	7A1	7A-1	15A1	15A-1
	3 P	1A	2A	2B	1A	1A	4A	4B	5A	2B	2A	7A-1	7A1	5A	5A
	5 P	1A	2A	2B	3A	3B	4A	4B	1A	6A	6B	7A-1	7A1	3A	3A
	7 P	1A	2A	2B	3A	3B	4A	4B	5A	6A	6B	1A	1A	15A-1	15A1
	Type
20160.a.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
20160.a.7a	R	$7$	$-1$	$3$	$4$	$1$	$-1$	$1$	$2$	$0$	$-1$	$0$	$0$	$-1$	$-1$
20160.a.14a	R	$14$	$6$	$2$	$-1$	$2$	$2$	$0$	$-1$	$-1$	$0$	$0$	$0$	$-1$	$-1$
20160.a.20a	R	$20$	$4$	$4$	$5$	$-1$	$0$	$0$	$0$	$1$	$1$	$-1$	$-1$	$0$	$0$
20160.a.21a	R	$21$	$-3$	$1$	$6$	$0$	$1$	$-1$	$1$	$-2$	$0$	$0$	$0$	$1$	$1$
20160.a.21b1	C	$21$	$-3$	$1$	$-3$	$0$	$1$	$-1$	$1$	$1$	$0$	$0$	$0$	$1-2{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3-2{\zeta }_{15}^4+{\zeta }_{15}^5-{\zeta }_{15}^7$	$-2+2{\zeta }_{15}+{\zeta }_{15}^2-{\zeta }_{15}^3+2{\zeta }_{15}^4-{\zeta }_{15}^5+{\zeta }_{15}^7$
20160.a.21b2	C	$21$	$-3$	$1$	$-3$	$0$	$1$	$-1$	$1$	$1$	$0$	$0$	$0$	$-2+2{\zeta }_{15}+{\zeta }_{15}^2-{\zeta }_{15}^3+2{\zeta }_{15}^4-{\zeta }_{15}^5+{\zeta }_{15}^7$	$1-2{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3-2{\zeta }_{15}^4+{\zeta }_{15}^5-{\zeta }_{15}^7$
20160.a.28a	R	$28$	$-4$	$4$	$1$	$1$	$0$	$0$	$-2$	$1$	$-1$	$0$	$0$	$1$	$1$
20160.a.35a	R	$35$	$3$	$-5$	$5$	$2$	$-1$	$-1$	$0$	$1$	$0$	$0$	$0$	$0$	$0$
20160.a.45a1	C	$45$	$-3$	$-3$	$0$	$0$	$1$	$1$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$0$	$0$
20160.a.45a2	C	$45$	$-3$	$-3$	$0$	$0$	$1$	$1$	$0$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	$0$	$0$
20160.a.56a	R	$56$	$8$	$0$	$-4$	$-1$	$0$	$0$	$1$	$0$	$-1$	$0$	$0$	$1$	$1$
20160.a.64a	R	$64$	$0$	$0$	$4$	$-2$	$0$	$0$	$-1$	$0$	$0$	$1$	$1$	$-1$	$-1$
20160.a.70a	R	$70$	$-2$	$2$	$-5$	$1$	$-2$	$0$	$0$	$-1$	$1$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed