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

## 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 4 3 2 4 3 3 2 1 1 3 . 1 4 1 3 2 1 2 1 1 1 1 1 . . . . 2 1 1 5 1 1 . . . . . . 1 1 . . . . . 7 1 . . . . . . . . . . 1 . . . 1a 2a 3a 6a 2b 6b 4a 12a 5a 10a 4b 7a 3b 2c 6c 2P 1a 1a 3a 3a 1a 3a 2b 6b 5a 5a 2b 7a 3b 1a 3b 3P 1a 2a 1a 2a 2b 2b 4a 4a 5a 10a 4b 7a 1a 2c 2c 5P 1a 2a 3a 6a 2b 6b 4a 12a 1a 2a 4b 7a 3b 2c 6c 7P 1a 2a 3a 6a 2b 6b 4a 12a 5a 10a 4b 1a 3b 2c 6c 11P 1a 2a 3a 6a 2b 6b 4a 12a 5a 10a 4b 7a 3b 2c 6c 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 3 1 2 -1 2 -1 1 -1 . -1 . . . X.4 6 -4 3 -1 2 -1 -2 1 1 1 . -1 . . . X.5 14 4 -1 1 2 -1 -2 1 -1 -1 . . 2 . . X.6 14 -6 2 . 2 2 . . -1 -1 . . -1 -2 1 X.7 14 -4 -1 -1 2 -1 2 -1 -1 1 . . 2 . . X.8 14 6 2 . 2 2 . . -1 1 . . -1 2 -1 X.9 15 -5 3 1 -1 -1 -1 -1 . . -1 1 . 3 . X.10 15 5 3 -1 -1 -1 1 1 . . -1 1 . -3 . X.11 20 . 2 . -4 2 . . . . . -1 2 . . X.12 21 1 -3 1 1 1 -1 -1 1 1 -1 . . -3 . X.13 21 -1 -3 -1 1 1 1 1 1 -1 -1 . . 3 . 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