Galois group: 35T44

• Label: 35T44
• Degree: $35$
• Order: $40320$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $S_8$

Group action invariants

Degree $n$:		$35$
Transitive number $t$:		$44$
Group:		$S_8$
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)(5,6)(9,12)(11,14)(17,23)(18,21)(19,24)(25,31)(27,29)(32,35)

Low degree resolvents

|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

8T50, 16T1838, 28T502, 30T1153

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
	$1^{35}$	$1$	$1$	$()$
	$2^{14},1^{7}$	$210$	$2$	$( 1,12)( 2,30)( 3, 9)( 4, 7)( 5,25)( 6,31)(10,15)(11,32)(13,28)(14,35)(17,18) (20,26)(21,23)(33,34)$
	$4^{7},2^{3},1$	$420$	$4$	$( 1,17,12,18)( 2,34,30,33)( 3,23, 9,21)( 4,15, 7,10)( 5,32,25,11)( 6,35,31,14) ( 8,16)(13,26,28,20)(19,29)(24,27)$
	$3^{10},1^{5}$	$112$	$3$	$( 1, 2, 3)( 4,25,31)( 5, 6, 7)( 8,24,19)( 9,12,30)(10,32,35)(11,14,15) (16,27,29)(17,34,23)(18,33,21)$
	$6^{4},3^{2},2^{2},1$	$1680$	$6$	$( 1, 9, 2,12, 3,30)( 4, 6,25, 7,31, 5)( 8,19,24)(10,14,32,15,35,11)(13,28) (16,29,27)(17,21,34,18,23,33)(20,26)$
	$12^{2},6,4,1$	$3360$	$12$	$( 1,34, 9,18, 2,23,12,33, 3,17,30,21)( 4,11, 6,10,25,14, 7,32,31,15, 5,35) ( 8,27,19,16,24,29)(13,26,28,20)$
	$4^{7},2^{3},1$	$2520$	$4$	$( 1,35,11,18)( 2, 6)( 3,34,14, 4)( 5,19)( 7,30,13,16)( 8,32,20,21) ( 9,27,17,31)(10,25,22,29)(12,28,23,33)(24,26)$
	$2^{12},1^{11}$	$105$	$2$	$( 1,13)( 2,26)( 3,20)( 4,32)( 6,24)( 7,11)( 8,14)( 9,17)(16,35)(18,30)(21,34) (27,31)$
	$2^{10},1^{15}$	$28$	$2$	$( 4,10)( 5,11)( 6,14)( 7,15)(17,18)(20,26)(21,23)(25,32)(31,35)(33,34)$
	$6^{3},3^{4},2,1^{3}$	$1120$	$6$	$( 2, 3,28)( 4,14,25,10, 6,32)( 5,15,31,11, 7,35)( 8,29,24)( 9,13,30)(16,19,27) (17,18)(20,33,23,26,34,21)$
	$5^{7}$	$1344$	$5$	$( 1,29, 6,19,17)( 2,16, 7, 8,34)( 3,27, 5,24,23)( 4,30,10,15,33) ( 9,32,11,21,25)(12,35,14,18,31)(13,26,20,22,28)$
	$15^{2},5$	$2688$	$15$	$( 1, 5,34,29,24, 2, 6,23,16,19, 3, 7,17,27, 8)( 4,35,21,30,14,25,10,18, 9,15, 31,32,33,12,11)(13,20,28,26,22)$
	$4^{6},2^{4},1^{3}$	$1260$	$4$	$( 1, 8,27,28)( 2, 7, 9,32)( 4,17)( 5,10)( 6,23,21,14)(11,26)(12,34,16,31) (13,35,24,33)(15,19,25,22)(18,20)$
	$8^{3},4^{2},2,1$	$5040$	$8$	$( 1,23, 8,21,27,14,28, 6)( 2,35, 7,24, 9,33,32,13)( 3,29)( 4,18,17,20) ( 5,11,10,26)(12,22,34,15,16,19,31,25)$
	$3^{11},1^{2}$	$1120$	$3$	$( 1, 9,16)( 2, 8,21)( 3, 4,24)( 5,19,15)( 6,20,32)( 7,31,30)(10,29,22) (11,27,18)(12,23,28)(13,17,35)(14,34,26)$
	$6^{3},3^{5},2$	$1120$	$6$	$( 1,16, 9)( 2, 4, 8,24,21, 3)( 5,15,19)( 6,34,20,26,32,14)( 7,30,31) (10,12,29,23,22,28)(11,18,27)(13,35,17)(25,33)$
	$2^{16},1^{3}$	$420$	$2$	$( 1,12)( 2,13)( 3, 9)( 4, 7)( 5, 6)(10,15)(11,14)(17,18)(19,27)(20,33)(21,23) (24,29)(25,31)(26,34)(28,30)(32,35)$
	$6^{5},3,2$	$3360$	$6$	$( 1,20, 2,13, 3,26)( 4,30,27,21,35,17)( 5,15,19)( 6, 7, 8,24,11,14) ( 9,32,18,31,34,16)(10,12,25,33,29,23)(22,28)$
	$6^{4},3^{3},1^{2}$	$3360$	$6$	$( 1,26, 3,13, 2,20)( 4,31,35,32,27,16)( 5,19,15)( 6,14,11,24, 8, 7) ( 9,30,34,17,18,21)(10,25,29)(12,33,23)$
	$10^{2},5^{3}$	$4032$	$10$	$( 1,19,28,24,27)( 2,16,29, 8, 3)( 4,23,33,15, 6,10,21,34, 7,14) ( 5,17,31,20,25,11,18,35,26,32)( 9,22,30,13,12)$
	$4^{7},2,1^{5}$	$1260$	$4$	$( 1,16,11,30)( 3, 4,14,34)( 5,19)( 7,18,13,35)( 8,21,20,32)( 9,31,17,27) (10,33,22,28)(12,29,23,25)$
	$7^{5}$	$5760$	$7$	$( 1,15,13,29, 8,20,28)( 2,18,30, 6,33,19,25)( 3,11,16,12,34,26,31) ( 4,22,35, 7,14,27,24)( 5,21, 9,23,10,17,32)$

Group invariants

Order:		$40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		40320.a
Character table:

	Size
	2 P
	3 P
	5 P
	7 P
	Type