Group action invariants
| Degree $n$ : | $35$ | |
| Transitive number $t$ : | $42$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,18,26,23,31,3,16,28,21,33)(2,17,27,22,32)(4,20,29,25,34,5,19,30,24,35)(6,13)(7,12)(8,11)(9,15)(10,14), (1,29,31,14,6,4,26,34,11,9)(2,28,32,13,7,3,27,33,12,8)(5,30,35,15,10)(16,24)(17,23)(18,22)(19,21)(20,25) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 10: $D_{5}$ 5040: $S_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 7: $S_7$
Low degree siblings
There are no siblings 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 $ | $2$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $525$ | $2$ | $( 2, 5)( 3, 4)( 6,11)( 7,15)( 8,14)( 9,13)(10,12)(16,21)(17,25)(18,24)(19,23) (20,22)(26,31)(27,35)(28,34)(29,33)(30,32)$ |
| $ 15, 15, 5 $ | $560$ | $15$ | $( 1, 4, 2, 5, 3)( 6,29,17,10,28,16, 9,27,20, 8,26,19, 7,30,18)(11,34,22,15,33, 21,14,32,25,13,31,24,12,35,23)$ |
| $ 15, 15, 5 $ | $560$ | $15$ | $( 1, 2, 3, 4, 5)( 6,27,18, 9,30,16, 7,28,19,10,26,17, 8,29,20)(11,32,23,14,35, 21,12,33,24,15,31,22,13,34,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $280$ | $3$ | $( 6,26,16)( 7,27,17)( 8,28,18)( 9,29,19)(10,30,20)(11,31,21)(12,32,22) (13,33,23)(14,34,24)(15,35,25)$ |
| $ 6, 6, 6, 6, 6, 2, 2, 1 $ | $4200$ | $6$ | $( 1, 4)( 2, 3)( 6,24,26,14,16,34)( 7,23,27,13,17,33)( 8,22,28,12,18,32) ( 9,21,29,11,19,31)(10,25,30,15,20,35)$ |
| $ 15, 5, 5, 5, 5 $ | $140$ | $15$ | $( 1, 4, 2, 5, 3)( 6,29,17,10,28,16, 9,27,20, 8,26,19, 7,30,18)(11,14,12,15,13) (21,24,22,25,23)(31,34,32,35,33)$ |
| $ 15, 5, 5, 5, 5 $ | $140$ | $15$ | $( 1, 2, 3, 4, 5)( 6,27,18, 9,30,16, 7,28,19,10,26,17, 8,29,20)(11,12,13,14,15) (21,22,23,24,25)(31,32,33,34,35)$ |
| $ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $70$ | $3$ | $( 6,26,16)( 7,27,17)( 8,28,18)( 9,29,19)(10,30,20)$ |
| $ 10, 10, 5, 5, 5 $ | $210$ | $10$ | $( 1,35, 4,33, 2,31, 5,34, 3,32)( 6,10, 9, 8, 7)(11,25,14,23,12,21,15,24,13,22) (16,20,19,18,17)(26,30,29,28,27)$ |
| $ 10, 10, 5, 5, 5 $ | $210$ | $10$ | $( 1,33, 5,32, 4,31, 3,35, 2,34)( 6, 8,10, 7, 9)(11,23,15,22,14,21,13,25,12,24) (16,18,20,17,19)(26,28,30,27,29)$ |
| $ 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 $ | $105$ | $2$ | $( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)(11,21)(12,22)(13,23)(14,24)(15,25)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $1050$ | $4$ | $( 1,24,31,14)( 2,23,32,13)( 3,22,33,12)( 4,21,34,11)( 5,25,35,15)( 6, 9) ( 7, 8)(16,19)(17,18)(26,29)(27,28)$ |
| $ 15, 10, 10 $ | $420$ | $30$ | $( 1,33, 5,32, 4,31, 3,35, 2,34)( 6,18,30, 7,19,26, 8,20,27, 9,16,28,10,17,29) (11,23,15,22,14,21,13,25,12,24)$ |
| $ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)( 6,16,26)( 7,17,27)( 8,18,28)( 9,19,29) (10,20,30)(11,21)(12,22)(13,23)(14,24)(15,25)$ |
| $ 15, 10, 10 $ | $420$ | $30$ | $( 1,35, 4,33, 2,31, 5,34, 3,32)( 6,20,29, 8,17,26,10,19,28, 7,16,30, 9,18,27) (11,25,14,23,12,21,15,24,13,22)$ |
| $ 6, 6, 4, 4, 4, 4, 4, 3 $ | $2100$ | $12$ | $( 1,13,31,23)( 2,12,32,22)( 3,11,33,21)( 4,15,34,25)( 5,14,35,24) ( 6,28,16, 8,26,18)( 7,27,17)( 9,30,19,10,29,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1, 4)( 2, 3)( 6,29)( 7,28)( 8,27)( 9,26)(10,30)(11,14)(12,13)(16,19)(17,18) (21,24)(22,23)(31,34)(32,33)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $630$ | $4$ | $( 1,21,31,11)( 2,22,32,12)( 3,23,33,13)( 4,24,34,14)( 5,25,35,15)( 6,26) ( 7,27)( 8,28)( 9,29)(10,30)$ |
| $ 20, 10, 5 $ | $1260$ | $20$ | $( 1,24,32,15, 3,21,34,12, 5,23,31,14, 2,25,33,11, 4,22,35,13)( 6,29, 7,30, 8, 26, 9,27,10,28)(16,19,17,20,18)$ |
| $ 20, 10, 5 $ | $1260$ | $20$ | $( 1,22,33,14, 5,21,32,13, 4,25,31,12, 3,24,35,11, 2,23,34,15)( 6,27, 8,29,10, 26, 7,28, 9,30)(16,17,18,19,20)$ |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $2100$ | $6$ | $( 1, 2)( 3, 5)( 6,12,16, 7,11,17)( 8,15,18,10,13,20)( 9,14,19)(21,27)(22,26) (23,30)(24,29)(25,28)(31,32)(33,35)$ |
| $ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $( 1,16, 6,31,11)( 2,17, 7,32,12)( 3,18, 8,33,13)( 4,19, 9,34,14) ( 5,20,10,35,15)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1008$ | $5$ | $( 1,19, 7,35,13)( 2,20, 8,31,14)( 3,16, 9,32,15)( 4,17,10,33,11) ( 5,18, 6,34,12)(21,24,22,25,23)(26,29,27,30,28)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1008$ | $5$ | $( 1,17, 8,34,15)( 2,18, 9,35,11)( 3,19,10,31,12)( 4,20, 6,32,13) ( 5,16, 7,33,14)(21,22,23,24,25)(26,27,28,29,30)$ |
| $ 10, 10, 5, 2, 2, 2, 2, 2 $ | $2520$ | $10$ | $( 1,12,31, 7,16, 2,11,32, 6,17)( 3,15,33,10,18, 5,13,35, 8,20)( 4,14,34, 9,19) (21,27)(22,26)(23,30)(24,29)(25,28)$ |
| $ 35 $ | $720$ | $35$ | $( 1,15,29,18, 7,31,25, 4,13,27,16,10,34,23, 2,11,30,19, 8,32,21, 5,14,28,17, 6,35,24, 3,12,26,20, 9,33,22)$ |
| $ 35 $ | $720$ | $35$ | $( 1,13,30,17, 9,31,23, 5,12,29,16, 8,35,22, 4,11,28,20, 7,34,21, 3,15,27,19, 6,33,25, 2,14,26,18,10,32,24)$ |
| $ 7, 7, 7, 7, 7 $ | $720$ | $7$ | $( 1,11,26,16, 6,31,21)( 2,12,27,17, 7,32,22)( 3,13,28,18, 8,33,23) ( 4,14,29,19, 9,34,24)( 5,15,30,20,10,35,25)$ |
| $ 35 $ | $720$ | $35$ | $( 1,12,28,19,10,31,22, 3,14,30,16, 7,33,24, 5,11,27,18, 9,35,21, 2,13,29,20, 6,32,23, 4,15,26,17, 8,34,25)$ |
| $ 35 $ | $720$ | $35$ | $( 1,14,27,20, 8,31,24, 2,15,28,16, 9,32,25, 3,11,29,17,10,33,21, 4,12,30,18, 6,34,22, 5,13,26,19, 7,35,23)$ |
Group invariants
| Order: | $25200=2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |