# Properties

 Label 35T31 Order $$5040$$ n $$35$$ 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) 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) $|\Aut(F/K)|$: $1$

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

## Group invariants

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