Group action invariants
| Degree $n$ : | $35$ | |
| Transitive number $t$ : | $19$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,3,2)(4,5)(6,35,8,34,7,31)(9,32)(10,33)(11,26)(12,27)(13,29,15,28,14,30)(16,21,20,25,19,24)(17,22)(18,23), (1,6,4,8)(2,9,3,7)(5,10)(11,31,13,34)(12,32)(14,35,15,33)(16,28,19,27)(17,29,20,30)(18,26)(21,23,25,24) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 14: $D_{7}$ 120: $S_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $S_5$
Degree 7: $D_{7}$
Low degree siblings
42T140Siblings are shown 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 $ | $1$ | $1$ | $()$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,35,28,24,20,13, 7)( 2,34,30,21,19,15, 6)( 3,31,29,25,16,14, 8) ( 4,33,27,23,17,11, 9)( 5,32,26,22,18,12,10)$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,24, 7,28,13,35,20)( 2,21, 6,30,15,34,19)( 3,25, 8,29,14,31,16) ( 4,23, 9,27,11,33,17)( 5,22,10,26,12,32,18)$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,28,20, 7,35,24,13)( 2,30,19, 6,34,21,15)( 3,29,16, 8,31,25,14) ( 4,27,17, 9,33,23,11)( 5,26,18,10,32,22,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $70$ | $2$ | $( 1, 8)( 2, 6)( 3, 7)( 4, 9)( 5,10)(11,33)(12,32)(13,31)(14,35)(15,34)(16,28) (17,27)(18,26)(19,30)(20,29)(24,25)$ |
| $ 14, 14, 7 $ | $30$ | $14$ | $( 1,14,24,31, 7,16,28, 3,13,25,35, 8,20,29)( 2,11,21,33, 6,17,30, 4,15,23,34, 9,19,27)( 5,12,22,32,10,18,26)$ |
| $ 14, 14, 7 $ | $30$ | $14$ | $( 1, 8,13,16,24,29,35, 3, 7,14,20,25,28,31)( 2, 9,15,17,21,27,34, 4, 6,11,19, 23,30,33)( 5,10,12,18,22,26,32)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 1, 3)( 2, 4)( 6, 9)( 7, 8)(11,15)(13,14)(16,20)(17,19)(21,23)(24,25)(27,30) (28,29)(31,35)(33,34)$ |
| $ 14, 14, 7 $ | $30$ | $14$ | $( 1,16,35,14,28, 8,24, 3,20,31,13,29, 7,25)( 2,17,34,11,30, 9,21, 4,19,33,15, 27, 6,23)( 5,18,32,12,26,10,22)$ |
| $ 21, 7, 7 $ | $40$ | $21$ | $( 1,14,21,35, 8,19,28, 3,15,24,31, 6,20,29, 2,13,25,34, 7,16,30) ( 4,11,23,33, 9,17,27)( 5,12,22,32,10,18,26)$ |
| $ 21, 7, 7 $ | $40$ | $21$ | $( 1, 8,15,20,25,30,35, 3, 6,13,16,21,28,31, 2, 7,14,19,24,29,34) ( 4, 9,11,17,23,27,33)( 5,10,12,18,22,26,32)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 1, 3, 2)( 6, 7, 8)(13,14,15)(16,19,20)(21,24,25)(28,29,30)(31,34,35)$ |
| $ 21, 7, 7 $ | $40$ | $21$ | $( 1,16,34,13,29, 6,24, 3,19,35,14,30, 7,25, 2,20,31,15,28, 8,21) ( 4,17,33,11,27, 9,23)( 5,18,32,12,26,10,22)$ |
| $ 6, 6, 6, 3, 2, 2, 2, 2, 2, 2, 2 $ | $140$ | $6$ | $( 1, 8, 2, 7, 3, 6)( 4,10)( 5, 9)(11,32)(12,33)(13,31,15,35,14,34) (16,30,20,29,19,28)(17,26)(18,27)(21,24,25)(22,23)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $210$ | $4$ | $( 1, 8, 2, 9)( 3, 6, 4, 7)( 5,10)(11,35,14,34)(12,32)(13,31,15,33) (16,30,17,28)(18,26)(19,27,20,29)(21,23,24,25)$ |
| $ 35 $ | $24$ | $35$ | $( 1,14,21,33,10,20,29, 2,11,22,35, 8,19,27, 5,13,25,34, 9,18,28, 3,15,23,32, 7,16,30, 4,12,24,31, 6,17,26)$ |
| $ 35 $ | $24$ | $35$ | $( 1, 8,15,17,22,28,31, 2, 9,12,20,25,30,33, 5, 7,14,19,23,26,35, 3, 6,11,18, 24,29,34, 4,10,13,16,21,27,32)$ |
| $ 35 $ | $24$ | $35$ | $( 1,31,30,23,18,13, 8, 2,33,26,24,16,15, 9, 5,35,29,21,17,12, 7, 3,34,27,22, 20,14, 6, 4,32,28,25,19,11,10)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $24$ | $5$ | $( 1, 3, 2, 4, 5)( 6, 9,10, 7, 8)(11,12,13,14,15)(16,19,17,18,20) (21,23,22,24,25)(26,28,29,30,27)(31,34,33,32,35)$ |
| $ 35 $ | $24$ | $35$ | $( 1,16,34,11,26, 7,25, 2,17,32,13,29, 6,23, 5,20,31,15,27,10,24, 3,19,33,12, 28, 8,21, 4,18,35,14,30, 9,22)$ |
| $ 35 $ | $24$ | $35$ | $( 1,29,19, 9,32,24,14, 2,27,18, 7,31,21,11, 5,28,16, 6,33,22,13, 3,30,17,10, 35,25,15, 4,26,20, 8,34,23,12)$ |
| $ 35 $ | $24$ | $35$ | $( 1,25, 6,27,12,35,16, 2,23,10,28,14,34,17, 5,24, 8,30,11,32,20, 3,21, 9,26, 13,31,19, 4,22, 7,29,15,33,18)$ |
Group invariants
| Order: | $840=2^{3} \cdot 3 \cdot 5 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | [840, 136] |
| Character table: Data not available. |