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

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.