Properties

Label 35T44
Order \(40320\)
n \(35\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No

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Group action invariants

Degree $n$ :  $35$
Transitive number $t$ :  $44$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

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

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

Group invariants

Order:  $40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.