Group action invariants
| Degree $n$: | $35$ | |
| Transitive number $t$: | $42$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| 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) |
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, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1,13)( 2,12)( 3,11)( 4,15)( 5,14)( 6, 8)( 9,10)(16,18)(19,20)(21,23)(24,25) (26,28)(29,30)(31,33)(34,35)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1008$ | $5$ | $( 1, 5, 4, 3, 2)( 6,25,19,28,32)( 7,21,20,29,33)( 8,22,16,30,34) ( 9,23,17,26,35)(10,24,18,27,31)(11,15,14,13,12)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1008$ | $5$ | $( 1, 3, 5, 2, 4)( 6,23,20,27,34)( 7,24,16,28,35)( 8,25,17,29,31) ( 9,21,18,30,32)(10,22,19,26,33)(11,13,15,12,14)$ |
| $ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $( 6,21,16,26,31)( 7,22,17,27,32)( 8,23,18,28,33)( 9,24,19,29,34) (10,25,20,30,35)$ |
| $ 10, 10, 5, 2, 2, 2, 2, 2 $ | $2520$ | $10$ | $( 1,13)( 2,12)( 3,11)( 4,15)( 5,14)( 6,28,21,33,16, 8,26,23,31,18) ( 7,27,22,32,17)( 9,30,24,35,19,10,29,25,34,20)$ |
| $ 10, 10, 5, 5, 5 $ | $210$ | $10$ | $( 1,29, 2,30, 3,26, 4,27, 5,28)( 6,19, 7,20, 8,16, 9,17,10,18)(11,14,12,15,13) (21,24,22,25,23)(31,34,32,35,33)$ |
| $ 10, 10, 5, 5, 5 $ | $210$ | $10$ | $( 1,27, 3,29, 5,26, 2,28, 4,30)( 6,17, 8,19,10,16, 7,18, 9,20)(11,12,13,14,15) (21,22,23,24,25)(31,32,33,34,35)$ |
| $ 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 $ | $105$ | $2$ | $( 1,26)( 2,27)( 3,28)( 4,29)( 5,30)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $630$ | $4$ | $( 1,16,26, 6)( 2,17,27, 7)( 3,18,28, 8)( 4,19,29, 9)( 5,20,30,10)(21,31) (22,32)(23,33)(24,34)(25,35)$ |
| $ 20, 10, 5 $ | $1260$ | $20$ | $( 1,19,27,10, 3,16,29, 7, 5,18,26, 9, 2,20,28, 6, 4,17,30, 8)(11,14,12,15,13) (21,34,22,35,23,31,24,32,25,33)$ |
| $ 20, 10, 5 $ | $1260$ | $20$ | $( 1,17,28, 9, 5,16,27, 8, 4,20,26, 7, 3,19,30, 6, 2,18,29,10)(11,12,13,14,15) (21,32,23,34,25,31,22,33,24,35)$ |
| $ 15, 5, 5, 5, 5 $ | $140$ | $15$ | $( 1, 3, 5, 2, 4)( 6,33,25, 7,34,21, 8,35,22, 9,31,23,10,32,24)(11,13,15,12,14) (16,18,20,17,19)(26,28,30,27,29)$ |
| $ 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 $ | $70$ | $3$ | $( 6,31,21)( 7,32,22)( 8,33,23)( 9,34,24)(10,35,25)$ |
| $ 15, 5, 5, 5, 5 $ | $140$ | $15$ | $( 1, 5, 4, 3, 2)( 6,35,24, 8,32,21,10,34,23, 7,31,25, 9,33,22)(11,15,14,13,12) (16,20,19,18,17)(26,30,29,28,27)$ |
| $ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,21,31)( 7,22,32)( 8,23,33)( 9,24,34) (10,25,35)(16,26)(17,27)(18,28)(19,29)(20,30)$ |
| $ 15, 10, 10 $ | $420$ | $30$ | $( 1,14, 2,15, 3,11, 4,12, 5,13)( 6,24,32,10,23,31, 9,22,35, 8,21,34, 7,25,33) (16,29,17,30,18,26,19,27,20,28)$ |
| $ 15, 10, 10 $ | $420$ | $30$ | $( 1,12, 3,14, 5,11, 2,13, 4,15)( 6,22,33, 9,25,31, 7,23,34,10,21,32, 8,24,35) (16,27,18,29,20,26,17,28,19,30)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $1050$ | $4$ | $( 1,29,11,19)( 2,28,12,18)( 3,27,13,17)( 4,26,14,16)( 5,30,15,20)( 6, 9) ( 7, 8)(21,24)(22,23)(31,34)(32,33)$ |
| $ 6, 6, 4, 4, 4, 4, 4, 3 $ | $2100$ | $12$ | $( 1,18,11,28)( 2,17,12,27)( 3,16,13,26)( 4,20,14,30)( 5,19,15,29) ( 6,23,31, 8,21,33)( 7,22,32)( 9,25,34,10,24,35)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $525$ | $2$ | $( 1,18)( 2,17)( 3,16)( 4,20)( 5,19)( 6, 8)( 9,10)(11,28)(12,27)(13,26)(14,30) (15,29)(21,33)(22,32)(23,31)(24,35)(25,34)$ |
| $ 15, 15, 5 $ | $560$ | $15$ | $( 1,35,29, 3,32,26, 5,34,28, 2,31,30, 4,33,27)( 6,10, 9, 8, 7)(11,20,24,13,17, 21,15,19,23,12,16,25,14,18,22)$ |
| $ 15, 15, 5 $ | $560$ | $15$ | $( 1,33,30, 2,34,26, 3,35,27, 4,31,28, 5,32,29)( 6, 8,10, 7, 9)(11,18,25,12,19, 21,13,20,22,14,16,23,15,17,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $280$ | $3$ | $( 1,31,26)( 2,32,27)( 3,33,28)( 4,34,29)( 5,35,30)(11,16,21)(12,17,22) (13,18,23)(14,19,24)(15,20,25)$ |
| $ 6, 6, 6, 6, 6, 2, 2, 1 $ | $4200$ | $6$ | $( 1,13,31,18,26,23)( 2,12,32,17,27,22)( 3,11,33,16,28,21)( 4,15,34,20,29,25) ( 5,14,35,19,30,24)( 6, 8)( 9,10)$ |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $2100$ | $6$ | $( 1, 2)( 3, 5)( 6,22,31, 7,21,32)( 8,25,33,10,23,35)( 9,24,34)(11,12)(13,15) (16,27)(17,26)(18,30)(19,29)(20,28)$ |
| $ 7, 7, 7, 7, 7 $ | $720$ | $7$ | $( 1, 6,21,31,16,11,26)( 2, 7,22,32,17,12,27)( 3, 8,23,33,18,13,28) ( 4, 9,24,34,19,14,29)( 5,10,25,35,20,15,30)$ |
| $ 35 $ | $720$ | $35$ | $( 1, 9,22,35,18,11,29, 2,10,23,31,19,12,30, 3, 6,24,32,20,13,26, 4, 7,25,33, 16,14,27, 5, 8,21,34,17,15,28)$ |
| $ 35 $ | $720$ | $35$ | $( 1, 7,23,34,20,11,27, 3, 9,25,31,17,13,29, 5, 6,22,33,19,15,26, 2, 8,24,35, 16,12,28, 4,10,21,32,18,14,30)$ |
| $ 35 $ | $720$ | $35$ | $( 1, 8,25,32,19,11,28, 5, 7,24,31,18,15,27, 4, 6,23,35,17,14,26, 3,10,22,34, 16,13,30, 2, 9,21,33,20,12,29)$ |
| $ 35 $ | $720$ | $35$ | $( 1,10,24,33,17,11,30, 4, 8,22,31,20,14,28, 2, 6,25,34,18,12,26, 5, 9,23,32, 16,15,29, 3, 7,21,35,19,13,27)$ |
Group invariants
| Order: | $25200=2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | no | |
| GAP id: | not available |
| Character table: not available. |