Group action invariants
| Degree $n$: | $35$ | |
| Transitive number $t$: | $50$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| 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) |
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 $ | $2016$ | $5$ | $( 1,23,10,17,29)( 2,24, 6,18,30)( 3,25, 7,19,26)( 4,21, 8,20,27) ( 5,22, 9,16,28)(11,13,15,12,14)(31,33,35,32,34)$ |
| $ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $( 1,21, 6,16,26)( 2,22, 7,17,27)( 3,23, 8,18,28)( 4,24, 9,19,29) ( 5,25,10,20,30)$ |
| $ 10, 10, 5, 2, 2, 2, 2, 1, 1 $ | $2520$ | $10$ | $( 1,24, 6,19,26, 4,21, 9,16,29)( 2,23, 7,18,27, 3,22, 8,17,28)( 5,25,10,20,30) (11,14)(12,13)(31,34)(32,33)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ | $105$ | $4$ | $( 1, 2, 5, 4)( 6, 7,10, 9)(11,32,15,34)(12,35,14,31)(13,33)(16,17,20,19) (21,22,25,24)(26,27,30,29)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ | $105$ | $4$ | $( 1, 5, 3, 4)( 6,10, 8, 9)(11,35,13,34)(12,32)(14,31,15,33)(16,20,18,19) (21,25,23,24)(26,30,28,29)$ |
| $ 20, 5, 4, 4, 2 $ | $2520$ | $20$ | $( 1, 8,29,22,16, 3, 9,27,21,18, 4, 7,26,23,19, 2, 6,28,24,17)( 5,10,30,25,20) (11,33,14,32)(12,31,13,34)(15,35)$ |
| $ 20, 5, 4, 4, 2 $ | $2520$ | $20$ | $( 1, 9,30,22,16, 4,10,27,21,19, 5, 7,26,24,20, 2, 6,29,25,17)( 3, 8,28,23,18) (11,34,15,32)(12,31,14,35)(13,33)$ |
| $ 10, 10, 5, 5, 5 $ | $420$ | $10$ | $( 1, 4, 2, 5, 3)( 6,29, 7,30, 8,26, 9,27,10,28)(11,19,12,20,13,16,14,17,15,18) (21,24,22,25,23)(31,34,32,35,33)$ |
| $ 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 $ | $105$ | $2$ | $( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(11,16)(12,17)(13,18)(14,19)(15,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $525$ | $2$ | $( 1, 3)( 4, 5)( 6,28)( 7,27)( 8,26)( 9,30)(10,29)(11,18)(12,17)(13,16)(14,20) (15,19)(21,23)(24,25)(31,33)(34,35)$ |
| $ 20, 10, 5 $ | $2520$ | $20$ | $( 1,23, 5,22, 4,21, 3,25, 2,24)( 6,13,30,17, 9,11,28,20, 7,14,26,18,10,12,29, 16, 8,15,27,19)(31,33,35,32,34)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $630$ | $4$ | $( 1,21)( 2,22)( 3,23)( 4,24)( 5,25)( 6,11,26,16)( 7,12,27,17)( 8,13,28,18) ( 9,14,29,19)(10,15,30,20)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ | $3150$ | $4$ | $( 1,24)( 2,23)( 3,22)( 4,21)( 5,25)( 6,14,26,19)( 7,13,27,18)( 8,12,28,17) ( 9,11,29,16)(10,15,30,20)(31,34)(32,33)$ |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $350$ | $6$ | $( 1,23,26, 3,21,28)( 2,22,27)( 4,25,29, 5,24,30)( 6, 8)( 9,10)(11,13)(14,15) (16,18)(19,20)(31,33)(34,35)$ |
| $ 15, 5, 5, 5, 5 $ | $280$ | $15$ | $( 1,24,27, 5,23,26, 4,22,30, 3,21,29, 2,25,28)( 6, 9, 7,10, 8)(11,14,12,15,13) (16,19,17,20,18)(31,34,32,35,33)$ |
| $ 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 $ | $70$ | $3$ | $( 1,21,26)( 2,22,27)( 3,23,28)( 4,24,29)( 5,25,30)$ |
| $ 12, 4, 4, 4, 4, 3, 2, 1, 1 $ | $2100$ | $12$ | $( 1,27,25, 4,26,22, 5,29,21, 2,30,24)( 3,28,23)( 6, 7,10, 9)(11,32,15,34) (12,35,14,31)(13,33)(16,17,20,19)$ |
| $ 12, 4, 4, 4, 4, 3, 2, 1, 1 $ | $2100$ | $12$ | $( 1,30,23, 4,26,25, 3,29,21, 5,28,24)( 2,27,22)( 6,10, 8, 9)(11,35,13,34) (12,32)(14,31,15,33)(16,20,18,19)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $1400$ | $6$ | $( 1,19,31, 4,16,34)( 2,18,32, 3,17,33)( 5,20,35)( 6,14,26, 9,11,29) ( 7,13,27, 8,12,28)(10,15,30)(21,24)(22,23)$ |
| $ 15, 15, 5 $ | $1120$ | $15$ | $( 1,18,35, 2,19,31, 3,20,32, 4,16,33, 5,17,34)( 6,13,30, 7,14,26, 8,15,27, 9, 11,28,10,12,29)(21,23,25,22,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $280$ | $3$ | $( 1,16,31)( 2,17,32)( 3,18,33)( 4,19,34)( 5,20,35)( 6,11,26)( 7,12,27) ( 8,13,28)( 9,14,29)(10,15,30)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $525$ | $4$ | $( 1,10, 2, 8)( 3, 6, 5, 7)( 4, 9)(11,20,12,18)(13,16,15,17)(14,19) (21,25,22,23)(26,35,27,33)(28,31,30,32)(29,34)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $525$ | $4$ | $( 1, 7, 4, 8)( 2, 9, 3, 6)( 5,10)(11,17,14,18)(12,19,13,16)(15,20) (21,22,24,23)(26,32,29,33)(27,34,28,31)(30,35)$ |
| $ 12, 12, 6, 4, 1 $ | $4200$ | $12$ | $( 1,29,35,12,16, 9, 5,27,31,14,20, 7)( 2,26,34,15,17, 6, 4,30,32,11,19,10) ( 3,28,33,13,18, 8)(21,24,25,22)$ |
| $ 12, 12, 6, 4, 1 $ | $4200$ | $12$ | $( 1,28,34,12,16, 8, 4,27,31,13,19, 7)( 2,26,33,14,17, 6, 3,29,32,11,18, 9) ( 5,30,35,15,20,10)(21,23,24,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $1050$ | $4$ | $( 1, 4, 5, 2)( 6,29,20,22)( 7,26,19,25)( 8,28,18,23)( 9,30,17,21)(10,27,16,24) (11,14,15,12)(31,34,35,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $1050$ | $4$ | $( 1, 3, 4, 2)( 6,28,19,22)( 7,26,18,24)( 8,29,17,21)( 9,27,16,23)(10,30,20,25) (11,13,14,12)(31,33,34,32)$ |
| $ 14, 14, 7 $ | $3600$ | $14$ | $( 1,31,16,26,11, 6,21)( 2,35,17,30,12,10,22, 5,32,20,27,15, 7,25) ( 3,34,18,29,13, 9,23, 4,33,19,28,14, 8,24)$ |
| $ 7, 7, 7, 7, 7 $ | $720$ | $7$ | $( 1,31,16,26,11, 6,21)( 2,32,17,27,12, 7,22)( 3,33,18,28,13, 8,23) ( 4,34,19,29,14, 9,24)( 5,35,20,30,15,10,25)$ |
| $ 35 $ | $1440$ | $35$ | $( 1,32,18,29,15, 6,22, 3,34,20,26,12, 8,24, 5,31,17,28,14,10,21, 2,33,19,30, 11, 7,23, 4,35,16,27,13, 9,25)$ |
| $ 35 $ | $1440$ | $35$ | $( 1,33,20,27,14, 6,23, 5,32,19,26,13,10,22, 4,31,18,30,12, 9,21, 3,35,17,29, 11, 8,25, 2,34,16,28,15, 7,24)$ |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1050$ | $6$ | $( 1,31,16)( 2,35,17, 5,32,20)( 3,34,18, 4,33,19)( 6,21)( 7,25)( 8,24)( 9,23) (10,22)(11,26)(12,30)(13,29)(14,28)(15,27)$ |
| $ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1,31,16)( 2,32,17)( 3,33,18)( 4,34,19)( 5,35,20)( 6,21)( 7,22)( 8,23)( 9,24) (10,25)(11,26)(12,27)(13,28)(14,29)(15,30)$ |
| $ 15, 10, 10 $ | $840$ | $30$ | $( 1,32,18, 4,35,16, 2,33,19, 5,31,17, 3,34,20)( 6,22, 8,24,10,21, 7,23, 9,25) (11,27,13,29,15,26,12,28,14,30)$ |
| $ 12, 4, 4, 4, 4, 4, 3 $ | $2100$ | $12$ | $( 1,20,32, 3,16,35, 2,18,31, 5,17,33)( 4,19,34)( 6,30,22,13)( 7,28,21,15) ( 8,26,25,12)( 9,29,24,14)(10,27,23,11)$ |
| $ 12, 4, 4, 4, 4, 4, 3 $ | $2100$ | $12$ | $( 1,17,34, 3,16,32, 4,18,31, 2,19,33)( 5,20,35)( 6,27,24,13)( 7,29,23,11) ( 8,26,22,14)( 9,28,21,12)(10,30,25,15)$ |
Group invariants
| Order: | $50400=2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | no | |
| GAP id: | not available |
| Character table: not available. |