Group action invariants
| Degree $n$ : | $35$ | |
| Transitive number $t$ : | $50$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,35,22,8,16,5,32,23,6,20,2,33,21,10,17,3,31,25,7,18)(4,34,24,9,19)(11,30,12,28)(13,26,15,27)(14,29), (1,8,20,2,9,16,3,10,17,4,6,18,5,7,19)(11,23,15,22,14,21,13,25,12,24)(26,33,30,32,29,31,28,35,27,34) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 4: $C_4$ 20: $F_5$ 5040: $S_7$ 10080: 28T360 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
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 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $2016$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,25,34,28,17)(12,21,35,29,18) (13,22,31,30,19)(14,23,32,26,20)(15,24,33,27,16)$ |
| $ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $(11,21,31,26,16)(12,22,32,27,17)(13,23,33,28,18)(14,24,34,29,19) (15,25,35,30,20)$ |
| $ 10, 10, 5, 2, 2, 2, 2, 1, 1 $ | $2520$ | $10$ | $( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,22,31,27,16,12,21,32,26,17)(13,25,33,30,18,15, 23,35,28,20)(14,24,34,29,19)$ |
| $ 14, 14, 7 $ | $3600$ | $14$ | $( 1,25,16,30,11,10,31, 5,21,20,26,15, 6,35)( 2,24,17,29,12, 9,32, 4,22,19,27, 14, 7,34)( 3,23,18,28,13, 8,33)$ |
| $ 35 $ | $1440$ | $35$ | $( 1,22,18,29,15, 6,32, 3,24,20,26,12, 8,34, 5,21,17,28,14,10,31, 2,23,19,30, 11, 7,33, 4,25,16,27,13, 9,35)$ |
| $ 35 $ | $1440$ | $35$ | $( 1,23,20,27,14, 6,33, 5,22,19,26,13,10,32, 4,21,18,30,12, 9,31, 3,25,17,29, 11, 8,35, 2,24,16,28,15, 7,34)$ |
| $ 7, 7, 7, 7, 7 $ | $720$ | $7$ | $( 1,21,16,26,11, 6,31)( 2,22,17,27,12, 7,32)( 3,23,18,28,13, 8,33) ( 4,24,19,29,14, 9,34)( 5,25,20,30,15,10,35)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $1400$ | $6$ | $( 1,31,16)( 2,35,17, 5,32,20)( 3,34,18, 4,33,19)( 7,10)( 8, 9)(11,26,21) (12,30,22,15,27,25)(13,29,23,14,28,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $280$ | $3$ | $( 1,31,16)( 2,32,17)( 3,33,18)( 4,34,19)( 5,35,20)(11,26,21)(12,27,22) (13,28,23)(14,29,24)(15,30,25)$ |
| $ 15, 15, 5 $ | $1120$ | $15$ | $( 1,32,18, 4,35,16, 2,33,19, 5,31,17, 3,34,20)( 6, 7, 8, 9,10)(11,27,23,14,30, 21,12,28,24,15,26,22,13,29,25)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $525$ | $4$ | $( 1,28, 4,27)( 2,26, 3,29)( 5,30)( 6, 8, 9, 7)(11,18,14,17)(12,16,13,19) (15,20)(21,33,24,32)(22,31,23,34)(25,35)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $525$ | $4$ | $( 1,29, 5,27)( 2,26, 4,30)( 3,28)( 6, 9,10, 7)(11,19,15,17)(12,16,14,20) (13,18)(21,34,25,32)(22,31,24,35)(23,33)$ |
| $ 12, 12, 6, 4, 1 $ | $4200$ | $12$ | $( 1,21,16,26,31,11)( 2,23,20,29,32,13, 5,24,17,28,35,14)( 3,25,19,27,33,15, 4, 22,18,30,34,12)( 7, 8,10, 9)$ |
| $ 12, 12, 6, 4, 1 $ | $4200$ | $12$ | $( 1,21,16,26,31,11)( 2,24,20,28,32,14, 5,23,17,29,35,13)( 3,22,19,30,33,12, 4, 25,18,27,34,15)( 7, 9,10, 8)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ | $105$ | $4$ | $( 1,15, 3,14)( 2,12)( 4,11, 5,13)( 6,10, 8, 9)(16,20,18,19)(21,25,23,24) (26,30,28,29)(31,35,33,34)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ | $105$ | $4$ | $( 1,12, 5,14)( 2,15, 4,11)( 3,13)( 6, 7,10, 9)(16,17,20,19)(21,22,25,24) (26,27,30,29)(31,32,35,34)$ |
| $ 20, 5, 4, 4, 2 $ | $2520$ | $20$ | $( 1,12, 4,13)( 2,14, 3,11)( 5,15)( 6,32,29,18,21, 7,34,28,16,22, 9,33,26,17, 24, 8,31,27,19,23)(10,35,30,20,25)$ |
| $ 20, 5, 4, 4, 2 $ | $2520$ | $20$ | $( 1,15, 2,13)( 3,11, 5,12)( 4,14)( 6,35,27,18,21,10,32,28,16,25, 7,33,26,20, 22, 8,31,30,17,23)( 9,34,29,19,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $525$ | $2$ | $( 1, 4)( 2, 3)( 6,24)( 7,23)( 8,22)( 9,21)(10,25)(11,34)(12,33)(13,32)(14,31) (15,35)(16,19)(17,18)(26,29)(27,28)$ |
| $ 10, 10, 5, 5, 5 $ | $420$ | $10$ | $( 1, 3, 5, 2, 4)( 6,23,10,22, 9,21, 8,25, 7,24)(11,33,15,32,14,31,13,35,12,34) (16,18,20,17,19)(26,28,30,27,29)$ |
| $ 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 $ | $105$ | $2$ | $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,31)(12,32)(13,33)(14,34)(15,35)$ |
| $ 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 $ | $70$ | $3$ | $( 1,26,16)( 2,27,17)( 3,28,18)( 4,29,19)( 5,30,20)$ |
| $ 15, 5, 5, 5, 5 $ | $280$ | $15$ | $( 1,27,18, 4,30,16, 2,28,19, 5,26,17, 3,29,20)( 6, 7, 8, 9,10)(11,12,13,14,15) (21,22,23,24,25)(31,32,33,34,35)$ |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $350$ | $6$ | $( 1,26,16)( 2,30,17, 5,27,20)( 3,29,18, 4,28,19)( 7,10)( 8, 9)(12,15)(13,14) (22,25)(23,24)(32,35)(33,34)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $1050$ | $4$ | $( 1, 4, 5, 2)( 6,14,25,32)( 7,11,24,35)( 8,13,23,33)( 9,15,22,31)(10,12,21,34) (16,19,20,17)(26,29,30,27)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $1050$ | $4$ | $( 1, 3, 4, 2)( 6,13,24,32)( 7,11,23,34)( 8,14,22,31)( 9,12,21,33)(10,15,25,35) (16,18,19,17)(26,28,29,27)$ |
| $ 15, 10, 10 $ | $840$ | $30$ | $( 1,17,28, 4,20,26, 2,18,29, 5,16,27, 3,19,30)( 6,22, 8,24,10,21, 7,23, 9,25) (11,32,13,34,15,31,12,33,14,35)$ |
| $ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1,16,26)( 2,17,27)( 3,18,28)( 4,19,29)( 5,20,30)( 6,21)( 7,22)( 8,23)( 9,24) (10,25)(11,31)(12,32)(13,33)(14,34)(15,35)$ |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1050$ | $6$ | $( 1,20,26, 5,16,30)( 2,19,27, 4,17,29)( 3,18,28)( 6,25)( 7,24)( 8,23)( 9,22) (10,21)(11,35)(12,34)(13,33)(14,32)(15,31)$ |
| $ 12, 4, 4, 4, 4, 4, 3 $ | $2100$ | $12$ | $( 1,28,17, 5,26,18, 2,30,16, 3,27,20)( 4,29,19)( 6,33,22,15)( 7,35,21,13) ( 8,32,25,11)( 9,34,24,14)(10,31,23,12)$ |
| $ 12, 4, 4, 4, 4, 4, 3 $ | $2100$ | $12$ | $( 1,29,18, 5,26,19, 3,30,16, 4,28,20)( 2,27,17)( 6,34,23,15)( 7,32,22,12) ( 8,35,21,14)( 9,33,25,11)(10,31,24,13)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ | $3150$ | $4$ | $( 1,11)( 2,15)( 3,14)( 4,13)( 5,12)( 6,16,26,21)( 7,20,27,25)( 8,19,28,24) ( 9,18,29,23)(10,17,30,22)(32,35)(33,34)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $630$ | $4$ | $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,16,26,21)( 7,17,27,22)( 8,18,28,23) ( 9,19,29,24)(10,20,30,25)$ |
| $ 20, 10, 5 $ | $2520$ | $20$ | $( 1,12, 3,14, 5,11, 2,13, 4,15)( 6,17,28,24,10,16,27,23, 9,20,26,22, 8,19,30, 21, 7,18,29,25)(31,32,33,34,35)$ |
| $ 12, 4, 4, 4, 4, 3, 2, 1, 1 $ | $2100$ | $12$ | $( 1,10,33, 4, 6,35, 3, 9,31, 5, 8,34)( 2, 7,32)(11,15,13,14)(16,20,18,19) (21,30,23,29)(22,27)(24,26,25,28)$ |
| $ 12, 4, 4, 4, 4, 3, 2, 1, 1 $ | $2100$ | $12$ | $( 1, 7,35, 4, 6,32, 5, 9,31, 2,10,34)( 3, 8,33)(11,12,15,14)(16,17,20,19) (21,27,25,29)(22,30,24,26)(23,28)$ |
Group invariants
| Order: | $50400=2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |