Group action invariants
| Degree $n$ : | $35$ | |
| Transitive number $t$ : | $40$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,33,11,3,31,13)(2,32,12)(4,35,14,5,34,15)(6,23)(7,22)(8,21)(9,25)(10,24)(16,28)(17,27)(18,26)(19,30)(20,29), (1,17,31,7)(2,16,32,6)(3,20,33,10)(4,19,34,9)(5,18,35,8)(11,12)(13,15)(21,27)(22,26)(23,30)(24,29)(25,28) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 10: $D_{5}$ 2520: $A_7$ 5040: $A_7\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 7: $A_7$
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, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $525$ | $2$ | $( 1,35)( 2,34)( 3,33)( 4,32)( 5,31)( 6,10)( 7, 9)(11,20)(12,19)(13,18)(14,17) (15,16)(21,25)(22,24)(26,30)(27,29)$ |
| $ 10, 10, 5, 5, 5 $ | $210$ | $10$ | $( 1,32, 3,34, 5,31, 2,33, 4,35)( 6, 7, 8, 9,10)(11,17,13,19,15,16,12,18,14,20) (21,22,23,24,25)(26,27,28,29,30)$ |
| $ 10, 10, 5, 5, 5 $ | $210$ | $10$ | $( 1,33, 5,32, 4,31, 3,35, 2,34)( 6, 8,10, 7, 9)(11,18,15,17,14,16,13,20,12,19) (21,23,25,22,24)(26,28,30,27,29)$ |
| $ 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 $ | $105$ | $2$ | $( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)(11,16)(12,17)(13,18)(14,19)(15,20)$ |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $350$ | $6$ | $( 1,35,16, 5,31,20)( 2,34,17, 4,32,19)( 3,33,18)( 6,10)( 7, 9)(11,15)(12,14) (21,25)(22,24)(26,30)(27,29)$ |
| $ 15, 5, 5, 5, 5 $ | $140$ | $15$ | $( 1,32,18, 4,35,16, 2,33,19, 5,31,17, 3,34,20)( 6, 7, 8, 9,10)(11,12,13,14,15) (21,22,23,24,25)(26,27,28,29,30)$ |
| $ 15, 5, 5, 5, 5 $ | $140$ | $15$ | $( 1,33,20, 2,34,16, 3,35,17, 4,31,18, 5,32,19)( 6, 8,10, 7, 9)(11,13,15,12,14) (21,23,25,22,24)(26,28,30,27,29)$ |
| $ 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 $ | $70$ | $3$ | $( 1,31,16)( 2,32,17)( 3,33,18)( 4,34,19)( 5,35,20)$ |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1050$ | $6$ | $( 1,35,16, 5,31,20)( 2,34,17, 4,32,19)( 3,33,18)( 6,15)( 7,14)( 8,13)( 9,12) (10,11)(21,30)(22,29)(23,28)(24,27)(25,26)$ |
| $ 15, 10, 10 $ | $420$ | $30$ | $( 1,32,18, 4,35,16, 2,33,19, 5,31,17, 3,34,20)( 6,12, 8,14,10,11, 7,13, 9,15) (21,27,23,29,25,26,22,28,24,30)$ |
| $ 15, 10, 10 $ | $420$ | $30$ | $( 1,33,20, 2,34,16, 3,35,17, 4,31,18, 5,32,19)( 6,13,10,12, 9,11, 8,15, 7,14) (21,28,25,27,24,26,23,30,22,29)$ |
| $ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1,31,16)( 2,32,17)( 3,33,18)( 4,34,19)( 5,35,20)( 6,11)( 7,12)( 8,13)( 9,14) (10,15)(21,26)(22,27)(23,28)(24,29)(25,30)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $1400$ | $6$ | $( 1,35,16, 5,31,20)( 2,34,17, 4,32,19)( 3,33,18)( 6,30,11,10,26,15) ( 7,29,12, 9,27,14)( 8,28,13)(21,25)(22,24)$ |
| $ 15, 15, 5 $ | $560$ | $15$ | $( 1,32,18, 4,35,16, 2,33,19, 5,31,17, 3,34,20)( 6,27,13, 9,30,11, 7,28,14,10, 26,12, 8,29,15)(21,22,23,24,25)$ |
| $ 15, 15, 5 $ | $560$ | $15$ | $( 1,33,20, 2,34,16, 3,35,17, 4,31,18, 5,32,19)( 6,28,15, 7,29,11, 8,30,12, 9, 26,13,10,27,14)(21,23,25,22,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $280$ | $3$ | $( 1,31,16)( 2,32,17)( 3,33,18)( 4,34,19)( 5,35,20)( 6,26,11)( 7,27,12) ( 8,28,13)( 9,29,14)(10,30,15)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ | $3150$ | $4$ | $( 1,35,16,15)( 2,34,17,14)( 3,33,18,13)( 4,32,19,12)( 5,31,20,11)( 6,30) ( 7,29)( 8,28)( 9,27)(10,26)(21,25)(22,24)$ |
| $ 20, 10, 5 $ | $1260$ | $20$ | $( 1,32,18,14, 5,31,17,13, 4,35,16,12, 3,34,20,11, 2,33,19,15)( 6,27, 8,29,10, 26, 7,28, 9,30)(21,22,23,24,25)$ |
| $ 20, 10, 5 $ | $1260$ | $20$ | $( 1,33,20,12, 4,31,18,15, 2,34,16,13, 5,32,19,11, 3,35,17,14)( 6,28,10,27, 9, 26, 8,30, 7,29)(21,23,25,22,24)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $630$ | $4$ | $( 1,31,16,11)( 2,32,17,12)( 3,33,18,13)( 4,34,19,14)( 5,35,20,15)( 6,26) ( 7,27)( 8,28)( 9,29)(10,30)$ |
| $ 10, 10, 5, 2, 2, 2, 2, 1, 1 $ | $2520$ | $10$ | $( 1,35,16,15, 6, 5,31,20,11,10)( 2,34,17,14, 7, 4,32,19,12, 9)( 3,33,18,13, 8) (21,25)(22,24)(26,30)(27,29)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1008$ | $5$ | $( 1,32,18,14,10)( 2,33,19,15, 6)( 3,34,20,11, 7)( 4,35,16,12, 8) ( 5,31,17,13, 9)(21,22,23,24,25)(26,27,28,29,30)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1008$ | $5$ | $( 1,33,20,12, 9)( 2,34,16,13,10)( 3,35,17,14, 6)( 4,31,18,15, 7) ( 5,32,19,11, 8)(21,23,25,22,24)(26,28,30,27,29)$ |
| $ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $( 1,31,16,11, 6)( 2,32,17,12, 7)( 3,33,18,13, 8)( 4,34,19,14, 9) ( 5,35,20,15,10)$ |
| $ 14, 14, 7 $ | $1800$ | $14$ | $( 1,35,16,15, 6,30,21, 5,31,20,11,10,26,25)( 2,34,17,14, 7,29,22, 4,32,19,12, 9,27,24)( 3,33,18,13, 8,28,23)$ |
| $ 35 $ | $720$ | $35$ | $( 1,32,18,14,10,26,22, 3,34,20,11, 7,28,24, 5,31,17,13, 9,30,21, 2,33,19,15, 6,27,23, 4,35,16,12, 8,29,25)$ |
| $ 35 $ | $720$ | $35$ | $( 1,33,20,12, 9,26,23, 5,32,19,11, 8,30,22, 4,31,18,15, 7,29,21, 3,35,17,14, 6,28,25, 2,34,16,13,10,27,24)$ |
| $ 7, 7, 7, 7, 7 $ | $360$ | $7$ | $( 1,31,16,11, 6,26,21)( 2,32,17,12, 7,27,22)( 3,33,18,13, 8,28,23) ( 4,34,19,14, 9,29,24)( 5,35,20,15,10,30,25)$ |
| $ 14, 14, 7 $ | $1800$ | $14$ | $( 1,35,16,15, 6,25,26, 5,31,20,11,10,21,30)( 2,34,17,14, 7,24,27, 4,32,19,12, 9,22,29)( 3,33,18,13, 8,23,28)$ |
| $ 35 $ | $720$ | $35$ | $( 1,32,18,14,10,21,27, 3,34,20,11, 7,23,29, 5,31,17,13, 9,25,26, 2,33,19,15, 6,22,28, 4,35,16,12, 8,24,30)$ |
| $ 35 $ | $720$ | $35$ | $( 1,33,20,12, 9,21,28, 5,32,19,11, 8,25,27, 4,31,18,15, 7,24,26, 3,35,17,14, 6,23,30, 2,34,16,13,10,22,29)$ |
| $ 7, 7, 7, 7, 7 $ | $360$ | $7$ | $( 1,31,16,11, 6,21,26)( 2,32,17,12, 7,22,27)( 3,33,18,13, 8,23,28) ( 4,34,19,14, 9,24,29)( 5,35,20,15,10,25,30)$ |
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. |