Properties

Label 35T44
Degree $35$
Order $40320$
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)
$|\Aut(F/K)|$:  $1$
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

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

Group invariants

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