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

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.