Group action invariants
| Degree $n$ : | $35$ | |
| Transitive number $t$ : | $44$ | |
| 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), (1,3)(5,6)(9,12)(11,14)(17,23)(18,21)(19,24)(25,31)(27,29)(32,35) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: None
Degree 7: None
Low degree siblings
8T50, 16T1838, 28T502, 30T1153Siblings 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 $ | $210$ | $2$ | $( 1,31)( 2,25)( 3, 4)( 5, 7)( 8,21)( 9,15)(10,32)(11,30)(12,14)(16,27)(18,19) (20,22)(23,34)(24,33)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $2520$ | $4$ | $( 1, 7,31, 5)( 2, 3,25, 4)( 6,28)( 8,30,21,11)( 9,24,15,33)(10,19,32,18) (12,16,14,27)(13,29)(20,34,22,23)(26,35)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1,20)( 2,25)( 3, 4)( 5,23)( 7,34)(10,27)(12,19)(13,35)(14,18)(16,32)(22,31) (26,29)$ |
| $ 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 $ | $28$ | $2$ | $( 8,15)( 9,21)(10,16)(11,24)(12,18)(13,26)(14,19)(27,32)(29,35)(30,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $1120$ | $3$ | $( 1, 5, 2)( 3,31,34)( 4,22, 7)( 6,28,17)( 8,21,30)( 9,33,15)(10,18,13) (12,26,16)(14,35,27)(19,29,32)(20,23,25)$ |
| $ 6, 6, 6, 3, 3, 3, 3, 3, 2 $ | $1120$ | $6$ | $( 1, 2, 5)( 3,34,31)( 4, 7,22)( 6,17,28)( 8,33,21,15,30, 9)(10,26,18,16,13,12) (11,24)(14,32,35,19,27,29)(20,25,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $112$ | $3$ | $( 1,22,28)( 4, 5,34)( 7,23,25)( 8, 9,27)(10,11,33)(12,13,29)(15,21,32) (16,24,30)(17,20,31)(18,26,35)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $420$ | $4$ | $( 1,11, 5,24)( 3,14, 6,19)( 4,16,28,10)( 7,20)( 8,15,13,26)( 9,21,29,35) (12,18,27,32)(17,25)(22,33,34,30)(23,31)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $1680$ | $6$ | $( 1,34,28, 5,22, 4)( 3, 6)( 7,23,25)( 8,29,27,13, 9,12)(10,24,33,16,11,30) (14,19)(15,35,32,26,21,18)(17,20,31)$ |
| $ 12, 12, 6, 4, 1 $ | $3360$ | $12$ | $( 1,16,34,11,28,30, 5,10,22,24, 4,33)( 3,19, 6,14)( 7,17,23,20,25,31) ( 8,18,29,15,27,35,13,32, 9,26,12,21)$ |
| $ 6, 6, 6, 3, 3, 3, 3, 2, 1, 1, 1 $ | $1120$ | $6$ | $( 1,28,22)( 4,34, 5)( 7,25,23)( 8,32, 9,15,27,21)(10,30,11,16,33,24) (12,35,13,18,29,26)(14,19)(17,31,20)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ | $1260$ | $4$ | $( 2,18, 3,12)( 4,19,25,14)( 5,35, 7,29)( 8,24,11,15)( 9,30,33,21)(10,16,27,32) (13,34,26,23)(17,28)$ |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1 $ | $1260$ | $4$ | $( 1,31,20,22)( 2,35,25,13)( 3,29, 4,26)( 5,19,23,12)( 7,14,34,18)( 8,21) ( 9,24)(10,32,27,16)(11,30)(15,33)$ |
| $ 8, 8, 8, 4, 4, 2, 1 $ | $5040$ | $8$ | $( 1,32,22,10,20,16,31,27)( 2,14,13, 7,25,18,35,34)( 3,12,26,23, 4,19,29, 5) ( 8,24,21, 9)(11,15,30,33)(17,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $420$ | $2$ | $( 1,22)( 2,12)( 3,18)( 4,14)( 5,13)( 7,26)( 8,33)( 9,11)(15,21)(16,32)(17,28) (19,25)(20,31)(23,35)(24,30)(29,34)$ |
| $ 7, 7, 7, 7, 7 $ | $5760$ | $7$ | $( 1,17,14, 2,22,20,11)( 3,34,13,24,18,31,10)( 4, 9, 8,27,29,16,21) ( 5,12,19,33,28,23,15)( 6,30,26,25,35, 7,32)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1344$ | $5$ | $( 1,29,33,25, 3)( 2,24,21,18, 6)( 4,19, 5,16,26)( 7,28, 8,31,27) ( 9,17,13,10,20)(11,23,22,12,14)(15,35,34,30,32)$ |
| $ 10, 10, 5, 5, 5 $ | $4032$ | $10$ | $( 1,25,29, 3,33)( 2,11,24,23,21,22,18,12, 6,14)( 4,13,19,10, 5,20,16, 9,26,17) ( 7,31,28,27, 8)(15,30,35,32,34)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 1, 1 $ | $3360$ | $6$ | $( 1, 9,16)( 2,35, 4,29, 5,23)( 3,19,18,12,13, 7)( 8,27,17,22,30,15) (10,28,31,21,24,11)(14,26,25)(20,33,32)$ |
| $ 6, 6, 6, 6, 6, 3, 2 $ | $3360$ | $6$ | $( 1,16, 9)( 2,23, 5,29, 4,35)( 3,17,13, 8,18,30)( 6,34)( 7,27,12,15,19,22) (10,11,24,21,31,28)(14,20,26,33,25,32)$ |
| $ 15, 15, 5 $ | $2688$ | $15$ | $( 1,10,31,23,19,22, 8,28, 7,35,17, 9,20, 3,26)( 2,33,29,16,30,34,27,13,15,32, 5,11,14,21,24)( 4,25, 6,18,12)$ |
Group invariants
| Order: | $40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |