Properties

Label 35T29
Order \(3360\)
n \(35\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Learn more about

Group action invariants

Degree $n$ :  $35$
Transitive number $t$ :  $29$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,30,32,23,6,20,12,3,26,35,22,8,16,15,2,28,31,25,7,18,11,5,27,33,21,10,17,13)(4,29,34,24,9,19,14), (1,7,35,24)(2,10,34,21)(3,8,33,23)(4,6,32,25)(5,9,31,22)(11,12,15,14)(16,27,20,29)(17,30,19,26)(18,28)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
20:  $F_5$
168:  $\GL(3,2)$
336:  14T17
672:  28T86

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Degree 7: $\GL(3,2)$

Low degree siblings

35T29, 40T2719

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $4$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$
$ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ $5$ $4$ $( 2, 3, 5, 4)( 7, 8,10, 9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29) (32,33,35,34)$
$ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ $5$ $4$ $( 2, 4, 5, 3)( 7, 9,10, 8)(12,14,15,13)(17,19,20,18)(22,24,25,23)(27,29,30,28) (32,34,35,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $105$ $2$ $( 2, 5)( 3, 4)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)(11,16)(12,20)(13,19)(14,18) (15,17)(22,25)(23,24)(27,30)(28,29)$
$ 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 $ $21$ $2$ $( 6,31)( 7,32)( 8,33)( 9,34)(10,35)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 10, 10, 5, 5, 5 $ $84$ $10$ $( 1, 5, 4, 3, 2)( 6,35, 9,33, 7,31,10,34, 8,32)(11,20,14,18,12,16,15,19,13,17) (21,25,24,23,22)(26,30,29,28,27)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ $105$ $4$ $( 2, 4, 5, 3)( 6,31)( 7,34,10,33)( 8,32, 9,35)(11,16)(12,19,15,18) (13,17,14,20)(22,24,25,23)(27,29,30,28)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ $105$ $4$ $( 2, 3, 5, 4)( 6,31)( 7,33,10,34)( 8,35, 9,32)(11,16)(12,18,15,19) (13,20,14,17)(22,23,25,24)(27,28,30,29)$
$ 12, 12, 4, 3, 3, 1 $ $280$ $12$ $( 1, 3, 4, 2)( 6,33,29, 7,31,28, 9,32,26, 8,34,27)(10,35,30)(11,23,19,12,21, 18,14,22,16,13,24,17)(15,25,20)$
$ 12, 12, 4, 3, 3, 1 $ $280$ $12$ $( 1, 4, 5, 2)( 6,34,30, 7,31,29,10,32,26, 9,35,27)( 8,33,28)(11,24,20,12,21, 19,15,22,16,14,25,17)(13,23,18)$
$ 15, 15, 5 $ $224$ $15$ $( 1, 5, 4, 3, 2)( 6,35,29, 8,32,26,10,34,28, 7,31,30, 9,33,27)(11,25,19,13,22, 16,15,24,18,12,21,20,14,23,17)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $56$ $3$ $( 6,31,26)( 7,32,27)( 8,33,28)( 9,34,29)(10,35,30)(11,21,16)(12,22,17) (13,23,18)(14,24,19)(15,25,20)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $280$ $6$ $( 1, 2)( 3, 5)( 6,32,26, 7,31,27)( 8,35,28,10,33,30)( 9,34,29)(11,22,16,12,21, 17)(13,25,18,15,23,20)(14,24,19)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ $210$ $4$ $( 1, 5, 3, 4)( 6,25,18,29)( 7,22,17,27)( 8,24,16,30)( 9,21,20,28)(10,23,19,26) (11,35,13,34)(12,32)(14,31,15,33)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ $210$ $4$ $( 1, 2, 5, 4)( 6,22,20,29)( 7,25,19,26)( 8,23,18,28)( 9,21,17,30)(10,24,16,27) (11,32,15,34)(12,35,14,31)(13,33)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ $210$ $4$ $( 1, 4)( 2, 3)( 6,24,16,29)( 7,23,17,28)( 8,22,18,27)( 9,21,19,26) (10,25,20,30)(11,34)(12,33)(13,32)(14,31)(15,35)$
$ 20, 10, 5 $ $168$ $20$ $( 1, 3, 5, 2, 4)( 6,23,20,27, 9,21,18,30, 7,24,16,28,10,22,19,26, 8,25,17,29) (11,33,15,32,14,31,13,35,12,34)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $42$ $4$ $( 6,21,16,26)( 7,22,17,27)( 8,23,18,28)( 9,24,19,29)(10,25,20,30)(11,31) (12,32)(13,33)(14,34)(15,35)$
$ 35 $ $96$ $35$ $( 1,28,10,17,14,21,33, 5,27, 9,16,13,25,32, 4,26, 8,20,12,24,31, 3,30, 7,19, 11,23,35, 2,29, 6,18,15,22,34)$
$ 7, 7, 7, 7, 7 $ $24$ $7$ $( 1,26, 6,16,11,21,31)( 2,27, 7,17,12,22,32)( 3,28, 8,18,13,23,33) ( 4,29, 9,19,14,24,34)( 5,30,10,20,15,25,35)$
$ 14, 14, 7 $ $120$ $14$ $( 1,29, 6,19,11,24,31, 4,26, 9,16,14,21,34)( 2,28, 7,18,12,23,32, 3,27, 8,17, 13,22,33)( 5,30,10,20,15,25,35)$
$ 28, 7 $ $120$ $28$ $( 1,30, 8,19,11,25,33, 4,26,10,18,14,21,35, 3,29, 6,20,13,24,31, 5,28, 9,16, 15,23,34)( 2,27, 7,17,12,22,32)$
$ 28, 7 $ $120$ $28$ $( 1,27,10,19,11,22,35, 4,26, 7,20,14,21,32, 5,29, 6,17,15,24,31, 2,30, 9,16, 12,25,34)( 3,28, 8,18,13,23,33)$
$ 14, 14, 7 $ $120$ $14$ $( 1,27, 6,22,31,12,16, 2,26, 7,21,32,11,17)( 3,30, 8,25,33,15,18, 5,28,10,23, 35,13,20)( 4,29, 9,24,34,14,19)$
$ 35 $ $96$ $35$ $( 1,30, 9,23,32,11,20, 4,28, 7,21,35,14,18, 2,26,10,24,33,12,16, 5,29, 8,22, 31,15,19, 3,27, 6,25,34,13,17)$
$ 7, 7, 7, 7, 7 $ $24$ $7$ $( 1,26, 6,21,31,11,16)( 2,27, 7,22,32,12,17)( 3,28, 8,23,33,13,18) ( 4,29, 9,24,34,14,19)( 5,30,10,25,35,15,20)$
$ 28, 7 $ $120$ $28$ $( 1,28, 9,22,31,13,19, 2,26, 8,24,32,11,18, 4,27, 6,23,34,12,16, 3,29, 7,21, 33,14,17)( 5,30,10,25,35,15,20)$
$ 28, 7 $ $120$ $28$ $( 1,29,10,22,31,14,20, 2,26, 9,25,32,11,19, 5,27, 6,24,35,12,16, 4,30, 7,21, 34,15,17)( 3,28, 8,23,33,13,18)$

Group invariants

Order:  $3360=2^{5} \cdot 3 \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.