# Properties

 Label 35T28 Degree $35$ Order $2520$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $A_7$

## Group action invariants

 Degree $n$: $35$ Transitive number $t$: $28$ Group: $A_7$ 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), (2,5,11)(3,7,15)(4,9,10)(6,13,14)(16,21,23)(17,18,22)(19,26,20)(25,30,32)(27,33,34)(28,31,35)

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 5: None

Degree 7: None

## Low degree siblings

7T6, 15T47 x 2, 21T33, 42T294, 42T299

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1$ $105$ $2$ $( 1,17)( 2,20)( 3,23)( 5,31)( 6,25)( 8,10)(11,19)(13,30)(14,24)(15,16)(26,33) (27,35)(28,34)(29,32)$ $4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1$ $630$ $4$ $( 1, 5,17,31)( 2,27,20,35)( 3,13,23,30)( 4,18)( 6,16,25,15)( 8,26,10,33) ( 9,22)(11,34,19,28)(12,21)(14,29,24,32)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1$ $70$ $3$ $( 1,17, 5)( 2,22,34)( 3,23, 6)( 4,28,20)( 8,10,26)( 9,35,19)(11,18,27) (12,32,24)(13,15,16)(14,21,29)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1$ $280$ $3$ $( 1,20, 9)( 2,26,27)( 3,24,13)( 4,35,17)( 5,28,19)( 6,32,16)( 7,25,30) ( 8,11,22)(10,18,34)(12,15,23)(14,21,29)$ $7, 7, 7, 7, 7$ $360$ $7$ $( 1, 9,20, 6,33,16,32)( 2,21,17,35, 8,13,25)( 3,22,23,34,29,27,28) ( 4,14,18,10,15,12,11)( 5,31,19,26, 7,30,24)$ $7, 7, 7, 7, 7$ $360$ $7$ $( 1,32,16,33, 6,20, 9)( 2,25,13, 8,35,17,21)( 3,28,27,29,34,23,22) ( 4,11,12,15,10,18,14)( 5,24,30, 7,26,19,31)$ $6, 6, 6, 6, 3, 3, 2, 2, 1$ $210$ $6$ $( 1, 7,17,10,12, 8)( 2,30,34)( 3,31,23,14, 9,29)( 4,24,15, 5,16,32)( 6,19,35) (11,13,27,28,25,20)(18,26)(22,33)$ $5, 5, 5, 5, 5, 5, 5$ $504$ $5$ $( 1,27, 2,17,33)( 3,16,14, 7,29)( 4,20,26,35, 9)( 5,34,22,18,11) ( 6,23,21,15,13)( 8,28,19,10,31)(12,25,24,32,30)$

## Group invariants

 Order: $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$ Cyclic: no Abelian: no Solvable: no GAP id: not available
 Character table:  2 3 . . . 3 2 2 . 2 3 2 . . . 1 2 1 2 . 5 1 1 . . . . . . . 7 1 . 1 1 . . . . . 1a 5a 7a 7b 2a 3a 6a 3b 4a 2P 1a 5a 7a 7b 1a 3a 3a 3b 2a 3P 1a 5a 7b 7a 2a 1a 2a 1a 4a 5P 1a 1a 7b 7a 2a 3a 6a 3b 4a 7P 1a 5a 1a 1a 2a 3a 6a 3b 4a X.1 1 1 1 1 1 1 1 1 1 X.2 6 1 -1 -1 2 3 -1 . . X.3 10 . A /A -2 1 1 1 . X.4 10 . /A A -2 1 1 1 . X.5 14 -1 . . 2 2 2 -1 . X.6 14 -1 . . 2 -1 -1 2 . X.7 15 . 1 1 -1 3 -1 . -1 X.8 21 1 . . 1 -3 1 . -1 X.9 35 . . . -1 -1 -1 -1 1 A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7