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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $1120$ | $3$ | $( 1,20,30)( 2, 8,13)( 3,26,16)( 4, 9, 5)( 6,10,27)( 7,28,19)(11,34,12) (14,33,29)(15,22,24)(17,25,35)(18,23,32)$ |
| $ 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 $ | $28$ | $2$ | $( 1,24)( 3,19)( 7,26)(15,20)(16,28)(17,32)(18,25)(21,31)(22,30)(23,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $112$ | $3$ | $( 1,17, 7)( 2,34, 6)( 3,22,23)( 8,12,10)(11,27,13)(14,33,29)(15,18,16) (19,30,35)(20,25,28)(24,32,26)$ |
| $ 6, 6, 6, 3, 3, 3, 3, 2, 1, 1, 1 $ | $1120$ | $6$ | $( 1,26,17,24, 7,32)( 2, 6,34)( 3,35,22,19,23,30)( 8,10,12)(11,13,27)(14,29,33) (15,28,18,20,16,25)(21,31)$ |
| $ 6, 6, 6, 3, 3, 3, 3, 3, 2 $ | $1120$ | $6$ | $( 1,22,20,24,30,15)( 2,13, 8)( 3,28,26,19,16, 7)( 4, 5, 9)( 6,27,10)(11,12,34) (14,29,33)(17,23,25,32,35,18)(21,31)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $420$ | $2$ | $( 1, 3)( 4,33)( 5,14)( 6,11)( 7,15)( 9,29)(10,34)(12,27)(16,30)(17,32)(18,25) (19,24)(20,26)(21,31)(22,28)(23,35)$ |
| $ 6, 6, 6, 6, 6, 3, 2 $ | $3360$ | $6$ | $( 1,26,30, 3,20,16)( 2, 8,13)( 4,29, 5,33, 9,14)( 6,34,27,11,10,12) ( 7,22,19,15,28,24)(17,18,35,32,25,23)(21,31)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1,16)( 3,30)( 4,11)( 5,10)( 6,33)( 8,13)(12,27)(14,34)(19,22)(21,35)(23,31) (24,28)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 1, 1 $ | $3360$ | $6$ | $( 1,22,21,16,19,35)( 2, 9,29)( 3,23,24,30,31,28)( 4,27, 6,11,12,33) ( 5,13,14,10, 8,34)( 7,20,32)(15,17,26)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $210$ | $2$ | $( 1,16)( 3,19)( 4, 5)( 7,18)( 8,27)(10,11)(12,13)(15,17)(20,32)(21,31)(22,30) (23,35)(24,28)(25,26)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $420$ | $4$ | $( 1,24,16,28)( 2,29)( 3, 8,19,27)( 4,31, 5,21)( 6,33)( 7,26,18,25) (10,35,11,23)(12,30,13,22)(14,34)(15,20,17,32)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $1680$ | $6$ | $( 1, 7,15,16,18,17)( 2,33,14)( 3,30,21,19,22,31)( 4,27,12, 5, 8,13)( 6,34,29) (10,11)(20,28,25,32,24,26)(23,35)$ |
| $ 12, 12, 6, 4, 1 $ | $3360$ | $12$ | $( 1,32, 7,24,15,26,16,20,18,28,17,25)( 2,34,33,29,14, 6)( 3, 5,30, 8,21,13,19, 4,22,27,31,12)(10,23,11,35)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $2520$ | $4$ | $( 1,21, 2, 9)( 3,22)( 4,11,31,24)( 6,13, 7,26)( 8,18,14,30)(10,35,29,16) (12,15,33,19)(17,32,34,27)(20,25)(23,28)$ |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1 $ | $1260$ | $4$ | $( 1,31,14,12)( 2, 9)( 3, 4)( 5,19,10,22)( 6, 8,28,21)( 7,20,32,18)(11,30) (13,24,35,33)(15,25)(16,23,34,27)$ |
| $ 8, 8, 8, 4, 4, 2, 1 $ | $5040$ | $8$ | $( 1,11,33,30,16, 4, 6, 3)( 5,14,22,24,10,34,19,28)( 7,15,18,25) ( 8,35,27,31,13,21,12,23)( 9,29)(17,20,26,32)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1344$ | $5$ | $( 1,18,19,14,29)( 2,33, 8,15,16)( 3,21,24,11,27)( 4,30,10,34, 7) ( 5,25, 9,32,23)( 6,31,12,35,17)(13,28,22,26,20)$ |
| $ 10, 10, 5, 5, 5 $ | $4032$ | $10$ | $( 1,11,18,27,19, 3,14,21,29,24)( 2,15,33,16, 8)( 4,34,30, 7,10) ( 5,35,25,17, 9, 6,32,31,23,12)(13,26,28,20,22)$ |
| $ 15, 15, 5 $ | $2688$ | $15$ | $( 1, 3, 9,31,28,22,21, 4,20,26,18,15, 8,19,35)( 2,23, 5,25,13,33,16,34,24, 6, 11, 7,27,30,29)(10,17,32,12,14)$ |
| $ 7, 7, 7, 7, 7 $ | $5760$ | $7$ | $( 1,17,16,28,31,19, 2)( 3, 5,18,10,29,20,30)( 4,35,14,24,22, 7,27) ( 6,11,12,23,13,25, 9)( 8,34,15,32,21,26,33)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ | $1260$ | $4$ | $( 1,15,16,17)( 2,34,29,14)( 3, 4,23,12)( 5,35,13,19)( 7,18)( 8,21,10,22) (11,30,27,31)(20,24,32,28)$ |
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. |