Group action invariants
| Degree $n$ : | $35$ | |
| Transitive number $t$ : | $20$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,4,5,2)(6,34,10,32)(7,31,9,35)(8,33)(11,29,15,27)(12,26,14,30)(13,28)(16,24,20,22)(17,21,19,25)(18,23), (1,22,13,19,35,6,2,23,14,20,31,7,3,24,15,16,32,8,4,25,11,17,33,9,5,21,12,18,34,10)(26,27,28,29,30) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_4$ x 2, $C_2^2$ 6: $C_6$ x 3 8: $C_4\times C_2$ 12: $C_{12}$ x 2, $C_6\times C_2$ 20: $F_5$ 24: 24T2 40: $F_{5}\times C_2$ 42: $F_7$ 60: $F_5\times C_3$ 84: $F_7 \times C_2$ 120: 30T26 168: 28T26 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Degree 7: $F_7$
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 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ |
| $ 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1,21, 6,26,11,31,16)( 2,22, 7,27,12,32,17)( 3,23, 8,28,13,33,18) ( 4,24, 9,29,14,34,19)( 5,25,10,30,15,35,20)$ |
| $ 35 $ | $24$ | $35$ | $( 1,22, 8,29,15,31,17, 3,24,10,26,12,33,19, 5,21, 7,28,14,35,16, 2,23, 9,30, 11,32,18, 4,25, 6,27,13,34,20)$ |
| $ 14, 14, 7 $ | $30$ | $14$ | $( 1,21, 6,26,11,31,16)( 2,25, 7,30,12,35,17, 5,22,10,27,15,32,20) ( 3,24, 8,29,13,34,18, 4,23, 9,28,14,33,19)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $35$ | $4$ | $( 2, 3, 5, 4)( 6,31)( 7,33,10,34)( 8,35, 9,32)(11,26)(12,28,15,29) (13,30,14,27)(16,21)(17,23,20,24)(18,25,19,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $35$ | $4$ | $( 2, 4, 5, 3)( 6,31)( 7,34,10,33)( 8,32, 9,35)(11,26)(12,29,15,28) (13,27,14,30)(16,21)(17,24,20,23)(18,22,19,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,31,26)(17,32,27) (18,33,28)(19,34,29)(20,35,30)$ |
| $ 15, 15, 5 $ | $28$ | $15$ | $( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,32,28,19,35, 26,17,33,29,20,31,27,18,34,30)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,11,21)( 7,15,22,10,12,25)( 8,14,23, 9,13,24)(16,31,26) (17,35,27,20,32,30)(18,34,28,19,33,29)$ |
| $ 12, 12, 6, 4, 1 $ | $35$ | $12$ | $( 2, 3, 5, 4)( 6,26,21,31,11,16)( 7,28,25,34,12,18,10,29,22,33,15,19) ( 8,30,24,32,13,20, 9,27,23,35,14,17)$ |
| $ 12, 12, 6, 4, 1 $ | $35$ | $12$ | $( 2, 4, 5, 3)( 6,26,21,31,11,16)( 7,29,25,33,12,19,10,28,22,34,15,18) ( 8,27,24,35,13,17, 9,30,23,32,14,20)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32) (18,28,33)(19,29,34)(20,30,35)$ |
| $ 15, 15, 5 $ | $28$ | $15$ | $( 1, 2, 3, 4, 5)( 6,22,13, 9,25,11, 7,23,14,10,21,12, 8,24,15)(16,27,33,19,30, 31,17,28,34,20,26,32,18,29,35)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,21,11)( 7,25,12,10,22,15)( 8,24,13, 9,23,14)(16,26,31) (17,30,32,20,27,35)(18,29,33,19,28,34)$ |
| $ 12, 12, 6, 4, 1 $ | $35$ | $12$ | $( 2, 3, 5, 4)( 6,16,11,31,21,26)( 7,18,15,34,22,28,10,19,12,33,25,29) ( 8,20,14,32,23,30, 9,17,13,35,24,27)$ |
| $ 12, 12, 6, 4, 1 $ | $35$ | $12$ | $( 2, 4, 5, 3)( 6,16,11,31,21,26)( 7,19,15,33,22,29,10,18,12,34,25,28) ( 8,17,14,35,23,27, 9,20,13,32,24,30)$ |
| $ 6, 6, 6, 6, 6, 1, 1, 1, 1, 1 $ | $7$ | $6$ | $( 6,16,11,31,21,26)( 7,17,12,32,22,27)( 8,18,13,33,23,28)( 9,19,14,34,24,29) (10,20,15,35,25,30)$ |
| $ 30, 5 $ | $28$ | $30$ | $( 1, 2, 3, 4, 5)( 6,17,13,34,25,26, 7,18,14,35,21,27, 8,19,15,31,22,28, 9,20, 11,32,23,29,10,16,12,33,24,30)$ |
| $ 6, 6, 6, 6, 6, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,16,11,31,21,26)( 7,20,12,35,22,30)( 8,19,13,34,23,29) ( 9,18,14,33,24,28)(10,17,15,32,25,27)$ |
| $ 12, 12, 4, 3, 3, 1 $ | $35$ | $12$ | $( 2, 3, 5, 4)( 6,21,11)( 7,23,15, 9,22,13,10,24,12, 8,25,14)(16,26,31) (17,28,35,19,27,33,20,29,32,18,30,34)$ |
| $ 12, 12, 4, 3, 3, 1 $ | $35$ | $12$ | $( 2, 4, 5, 3)( 6,21,11)( 7,24,15, 8,22,14,10,23,12, 9,25,13)(16,26,31) (17,29,35,18,27,34,20,28,32,19,30,33)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 6,31)( 7,32)( 8,33)( 9,34)(10,35)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21) (17,22)(18,23)(19,24)(20,25)$ |
| $ 10, 10, 10, 5 $ | $28$ | $10$ | $( 1, 2, 3, 4, 5)( 6,32, 8,34,10,31, 7,33, 9,35)(11,27,13,29,15,26,12,28,14,30) (16,22,18,24,20,21,17,23,19,25)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $35$ | $2$ | $( 2, 5)( 3, 4)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)(11,26)(12,30)(13,29)(14,28) (15,27)(16,21)(17,25)(18,24)(19,23)(20,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 2, 3, 5, 4)( 7, 8,10, 9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29) (32,33,35,34)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 2, 4, 5, 3)( 7, 9,10, 8)(12,14,15,13)(17,19,20,18)(22,24,25,23)(27,29,30,28) (32,34,35,33)$ |
| $ 28, 7 $ | $30$ | $28$ | $( 1,21, 6,26,11,31,16)( 2,23,10,29,12,33,20, 4,22, 8,30,14,32,18, 5,24, 7,28, 15,34,17, 3,25, 9,27,13,35,19)$ |
| $ 28, 7 $ | $30$ | $28$ | $( 1,21, 6,26,11,31,16)( 2,24,10,28,12,34,20, 3,22, 9,30,13,32,19, 5,23, 7,29, 15,33,17, 4,25, 8,27,14,35,18)$ |
| $ 6, 6, 6, 6, 6, 1, 1, 1, 1, 1 $ | $7$ | $6$ | $( 6,26,21,31,11,16)( 7,27,22,32,12,17)( 8,28,23,33,13,18)( 9,29,24,34,14,19) (10,30,25,35,15,20)$ |
| $ 30, 5 $ | $28$ | $30$ | $( 1, 2, 3, 4, 5)( 6,27,23,34,15,16, 7,28,24,35,11,17, 8,29,25,31,12,18, 9,30, 21,32,13,19,10,26,22,33,14,20)$ |
| $ 6, 6, 6, 6, 6, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,26,21,31,11,16)( 7,30,22,35,12,20)( 8,29,23,34,13,19) ( 9,28,24,33,14,18)(10,27,25,32,15,17)$ |
| $ 12, 12, 4, 3, 3, 1 $ | $35$ | $12$ | $( 2, 3, 5, 4)( 6,11,21)( 7,13,25, 9,12,23,10,14,22, 8,15,24)(16,31,26) (17,33,30,19,32,28,20,34,27,18,35,29)$ |
| $ 12, 12, 4, 3, 3, 1 $ | $35$ | $12$ | $( 2, 4, 5, 3)( 6,11,21)( 7,14,25, 8,12,24,10,13,22, 9,15,23)(16,31,26) (17,34,30,18,32,29,20,33,27,19,35,28)$ |
Group invariants
| Order: | $840=2^{3} \cdot 3 \cdot 5 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [840, 139] |
| Character table: Data not available. |