# Properties

 Label 35T31 Degree $35$ Order $5040$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no

## Group action invariants

 Degree $n$: $35$ Transitive number $t$: $31$ 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), (2,5)(3,7)(9,10)(13,14)(16,21)(17,22)(19,26)(27,33)(28,31)(30,32)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Degree 5: None

Degree 7: None

## Low degree siblings

7T7, 14T46, 21T38, 30T565, 42T411, 42T412, 42T413, 42T418

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 5, 5, 5, 5, 5$ $504$ $5$ $( 1,18,25,30, 2)( 3,15, 4,31,19)( 5,17,27,32,34)( 6, 7,16,28,35) ( 8,21,20, 9,14)(10,29,26,23,13)(11,12,33,24,22)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $21$ $2$ $( 1,31)( 2, 4)( 3,25)( 8,33)( 9,11)(12,14)(15,30)(18,19)(20,22)(21,24)$ $10, 10, 5, 5, 5$ $504$ $10$ $( 1, 3, 2,19,30,31,25, 4,18,15)( 5,27,34,17,32)( 6,16,35, 7,28) ( 8,22,14,24, 9,33,20,12,21,11)(10,26,13,29,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $105$ $2$ $( 1,28)( 2,22)( 4,12)( 5,13)( 6, 9)( 7,27)( 8,25)(10,18)(11,26)(14,21)(15,32) (16,17)(19,23)(20,24)(29,31)(30,34)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1$ $280$ $3$ $( 1,14,30)( 2,31,15)( 3,33,35)( 5,20,16)( 6,25,18)( 7,11,19)( 8,10, 9) (13,24,17)(21,34,28)(22,29,32)(23,27,26)$ $6, 6, 6, 6, 6, 3, 2$ $840$ $6$ $( 1,34,14,28,30,21)( 2,32,31,22,15,29)( 3,35,33)( 4,12)( 5,17,20,13,16,24) ( 6,10,25, 9,18, 8)( 7,23,11,27,19,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1$ $70$ $3$ $( 4,10,16)( 5,11,24)( 6,14,19)( 7,15, 8)( 9,23,21)(12,17,18)(13,20,26) (25,32,27)(29,31,35)(30,34,33)$ $6, 6, 6, 3, 3, 3, 3, 2, 1, 1, 1$ $420$ $6$ $( 1, 2)( 4,29,10,31,16,35)( 5,24,11)( 6, 8,14, 7,19,15)( 9,21,23) (12,33,17,30,18,34)(13,26,20)(25,27,32)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1$ $105$ $2$ $( 1, 2)( 4,17)( 6, 7)( 8,19)( 9,27)(10,18)(12,16)(14,15)(21,32)(22,28)(23,25) (29,30)(31,34)(33,35)$ $4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1$ $210$ $4$ $( 1,28, 2,22)( 4,34,17,31)( 5,20)( 6,23, 7,25)( 8,27,19, 9)(10,33,18,35) (11,26)(12,29,16,30)(13,24)(14,21,15,32)$ $6, 6, 6, 6, 3, 3, 2, 2, 1$ $210$ $6$ $( 1, 2)( 4,12,10,17,16,18)( 5,24,11)( 6, 8,14, 7,19,15)( 9,32,23,27,21,25) (13,26,20)(22,28)(29,33,31,30,35,34)$ $12, 12, 6, 4, 1$ $420$ $12$ $( 1, 2,28,22)( 4,13,18, 5,16,26,17,24,10,20,12,11)( 6,27,21, 7,19,32,23, 8,14, 25, 9,15)(29,33,31,30,35,34)$ $7, 7, 7, 7, 7$ $720$ $7$ $( 1, 6,10,21,30,20,27)( 2,14,33,31,25,28, 3)( 4,32,22,19,11,29, 7) ( 5,12,26, 9,24,13, 8)(15,34,17,18,35,23,16)$ $4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1$ $630$ $4$ $( 1,19,23,32)( 2,20,10,15)( 3,24,17,35)( 5,31, 9,21)( 6,25,12,18)( 7,30,22,26) ( 8,16,34,28)(11,14)(13,33)(27,29)$

## Group invariants

 Order: $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ Cyclic: no Abelian: no Solvable: no GAP id: not available
 Character table:  2 4 4 1 1 4 3 3 2 . 4 1 1 3 3 2 3 2 1 . . 1 . 2 1 . 1 2 1 1 1 1 5 1 1 1 1 . . . . . . . . . . . 7 1 . . . . . . . 1 . . . . . . 1a 2a 5a 10a 2b 4a 3a 6a 7a 2c 3b 6b 4b 6c 12a 2P 1a 1a 5a 5a 1a 2b 3a 3a 7a 1a 3b 3b 2b 3a 6c 3P 1a 2a 5a 10a 2b 4a 1a 2a 7a 2c 1a 2c 4b 2b 4b 5P 1a 2a 1a 2a 2b 4a 3a 6a 7a 2c 3b 6b 4b 6c 12a 7P 1a 2a 5a 10a 2b 4a 3a 6a 1a 2c 3b 6b 4b 6c 12a 11P 1a 2a 5a 10a 2b 4a 3a 6a 7a 2c 3b 6b 4b 6c 12a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 -1 1 -1 X.3 6 4 1 -1 2 . 3 1 -1 . . . 2 -1 -1 X.4 6 -4 1 1 2 . 3 -1 -1 . . . -2 -1 1 X.5 14 4 -1 -1 2 . -1 1 . . 2 . -2 -1 1 X.6 14 -6 -1 -1 2 . 2 . . -2 -1 1 . 2 . X.7 14 -4 -1 1 2 . -1 -1 . . 2 . 2 -1 -1 X.8 14 6 -1 1 2 . 2 . . 2 -1 -1 . 2 . X.9 15 -5 . . -1 -1 3 1 1 3 . . -1 -1 -1 X.10 15 5 . . -1 -1 3 -1 1 -3 . . 1 -1 1 X.11 20 . . . -4 . 2 . -1 . 2 . . 2 . X.12 21 1 1 1 1 -1 -3 1 . -3 . . -1 1 -1 X.13 21 -1 1 -1 1 -1 -3 -1 . 3 . . 1 1 1 X.14 35 -5 . . -1 1 -1 1 . -1 -1 -1 1 -1 1 X.15 35 5 . . -1 1 -1 -1 . 1 -1 1 -1 -1 -1