Group action invariants
| Degree $n$ : | $34$ | |
| Transitive number $t$ : | $27$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,25,34,2,26,33)(3,12,5,4,11,6)(7,21,24,8,22,23)(9,10)(13,16,31,14,15,32)(17,18)(19,28,30,20,27,29), (1,28,3,15,32,33,14,29,10,18,11,7,26,22,23,20,5,2,27,4,16,31,34,13,30,9,17,12,8,25,21,24,19,6) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 4080: $\PSL(2,16)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $\PSL(2,16)$
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)$ |
| $ 17, 17 $ | $240$ | $17$ | $( 1,30,11,26,17,16,32,34, 5,21,19,27,10,23, 3,14, 8)( 2,29,12,25,18,15,31,33, 6,22,20,28, 9,24, 4,13, 7)$ |
| $ 34 $ | $240$ | $34$ | $( 1,29,11,25,17,15,32,33, 5,22,19,28,10,24, 3,13, 8, 2,30,12,26,18,16,31,34, 6,21,20,27, 9,23, 4,14, 7)$ |
| $ 17, 17 $ | $240$ | $17$ | $( 1,17, 5,10, 8,26,34,27,14,11,32,19, 3,30,16,21,23)( 2,18, 6, 9, 7,25,33,28, 13,12,31,20, 4,29,15,22,24)$ |
| $ 34 $ | $240$ | $34$ | $( 1,18, 5, 9, 8,25,34,28,14,12,32,20, 3,29,16,22,23, 2,17, 6,10, 7,26,33,27, 13,11,31,19, 4,30,15,21,24)$ |
| $ 34 $ | $240$ | $34$ | $( 1,12,17,31, 5,20,10, 4, 8,29,26,15,34,22,27,24,14, 2,11,18,32, 6,19, 9, 3, 7,30,25,16,33,21,28,23,13)$ |
| $ 17, 17 $ | $240$ | $17$ | $( 1,11,17,32, 5,19,10, 3, 8,30,26,16,34,21,27,23,14)( 2,12,18,31, 6,20, 9, 4, 7,29,25,15,33,22,28,24,13)$ |
| $ 34 $ | $240$ | $34$ | $( 1, 6, 8,33,14,31, 3,15,23,18,10,25,27,12,19,29,21, 2, 5, 7,34,13,32, 4,16, 24,17, 9,26,28,11,20,30,22)$ |
| $ 17, 17 $ | $240$ | $17$ | $( 1, 5, 8,34,14,32, 3,16,23,17,10,26,27,11,19,30,21)( 2, 6, 7,33,13,31, 4,15, 24,18, 9,25,28,12,20,29,22)$ |
| $ 34 $ | $240$ | $34$ | $( 1,31,10,29,34,24,11, 6, 3,25,21,13,17,20, 8,15,27, 2,32, 9,30,33,23,12, 5, 4,26,22,14,18,19, 7,16,28)$ |
| $ 17, 17 $ | $240$ | $17$ | $( 1,32,10,30,34,23,11, 5, 3,26,21,14,17,19, 8,16,27)( 2,31, 9,29,33,24,12, 6, 4,25,22,13,18,20, 7,15,28)$ |
| $ 17, 17 $ | $240$ | $17$ | $( 1,34, 3,17,27,30, 5,14,16,10,11,21, 8,32,23,26,19)( 2,33, 4,18,28,29, 6,13, 15, 9,12,22, 7,31,24,25,20)$ |
| $ 34 $ | $240$ | $34$ | $( 1,33, 3,18,27,29, 5,13,16, 9,11,22, 8,31,23,25,19, 2,34, 4,17,28,30, 6,14, 15,10,12,21, 7,32,24,26,20)$ |
| $ 34 $ | $240$ | $34$ | $( 1, 9,34,12, 3,22,17, 7,27,31,30,24, 5,25,14,20,16, 2,10,33,11, 4,21,18, 8, 28,32,29,23, 6,26,13,19,15)$ |
| $ 17, 17 $ | $240$ | $17$ | $( 1,10,34,11, 3,21,17, 8,27,32,30,23, 5,26,14,19,16)( 2, 9,33,12, 4,22,18, 7, 28,31,29,24, 6,25,13,20,15)$ |
| $ 34 $ | $240$ | $34$ | $( 1, 4,27, 6,16,12, 8,24,19,33,17,29,14, 9,21,31,26, 2, 3,28, 5,15,11, 7,23, 20,34,18,30,13,10,22,32,25)$ |
| $ 17, 17 $ | $240$ | $17$ | $( 1, 3,27, 5,16,11, 8,23,19,34,17,30,14,10,21,32,26)( 2, 4,28, 6,15,12, 7,24, 20,33,18,29,13, 9,22,31,25)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $272$ | $5$ | $( 1, 8,11,17,16)( 2, 7,12,18,15)( 3,26,30,19, 5)( 4,25,29,20, 6) ( 9,28,31,33,22)(10,27,32,34,21)$ |
| $ 10, 10, 10, 2, 2 $ | $272$ | $10$ | $( 1, 7,11,18,16, 2, 8,12,17,15)( 3,25,30,20, 5, 4,26,29,19, 6)( 9,27,31,34,22, 10,28,32,33,21)(13,14)(23,24)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $272$ | $5$ | $( 1,17, 8,16,11)( 2,18, 7,15,12)( 3,19,26, 5,30)( 4,20,25, 6,29) ( 9,33,28,22,31)(10,34,27,21,32)$ |
| $ 10, 10, 10, 2, 2 $ | $272$ | $10$ | $( 1,18, 8,15,11, 2,17, 7,16,12)( 3,20,26, 6,30, 4,19,25, 5,29)( 9,34,28,21,31, 10,33,27,22,32)(13,14)(23,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $272$ | $3$ | $( 1,30,10)( 2,29, 9)( 3,34,17)( 4,33,18)( 5,32,11)( 6,31,12)( 7,20,28) ( 8,19,27)(15,25,22)(16,26,21)$ |
| $ 6, 6, 6, 6, 6, 2, 2 $ | $272$ | $6$ | $( 1,29,10, 2,30, 9)( 3,33,17, 4,34,18)( 5,31,11, 6,32,12)( 7,19,28, 8,20,27) (13,14)(15,26,22,16,25,21)(23,24)$ |
| $ 15, 15, 1, 1, 1, 1 $ | $272$ | $15$ | $( 1,26,34,11,19,10,16, 3,32, 8,30,21,17, 5,27)( 2,25,33,12,20, 9,15, 4,31, 7, 29,22,18, 6,28)$ |
| $ 30, 2, 2 $ | $272$ | $30$ | $( 1,25,34,12,19, 9,16, 4,32, 7,30,22,17, 6,27, 2,26,33,11,20,10,15, 3,31, 8, 29,21,18, 5,28)(13,14)(23,24)$ |
| $ 15, 15, 1, 1, 1, 1 $ | $272$ | $15$ | $( 1,21, 3,11,27,30,16,34, 5, 8,10,26,17,32,19)( 2,22, 4,12,28,29,15,33, 6, 7, 9,25,18,31,20)$ |
| $ 30, 2, 2 $ | $272$ | $30$ | $( 1,22, 3,12,27,29,16,33, 5, 7,10,25,17,31,19, 2,21, 4,11,28,30,15,34, 6, 8, 9,26,18,32,20)(13,14)(23,24)$ |
| $ 15, 15, 1, 1, 1, 1 $ | $272$ | $15$ | $( 1, 3,27,16, 5,10,17,19,21,11,30,34, 8,26,32)( 2, 4,28,15, 6, 9,18,20,22,12, 29,33, 7,25,31)$ |
| $ 30, 2, 2 $ | $272$ | $30$ | $( 1, 4,27,15, 5, 9,17,20,21,12,30,33, 8,25,32, 2, 3,28,16, 6,10,18,19,22,11, 29,34, 7,26,31)(13,14)(23,24)$ |
| $ 15, 15, 1, 1, 1, 1 $ | $272$ | $15$ | $( 1,34,19,16,32,30,17,27,26,11,10, 3, 8,21, 5)( 2,33,20,15,31,29,18,28,25,12, 9, 4, 7,22, 6)$ |
| $ 30, 2, 2 $ | $272$ | $30$ | $( 1,33,19,15,32,29,17,28,26,12,10, 4, 8,22, 5, 2,34,20,16,31,30,18,27,25,11, 9, 3, 7,21, 6)(13,14)(23,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $255$ | $2$ | $( 1, 6)( 2, 5)( 3,29)( 4,30)( 7,10)( 8, 9)(11,33)(12,34)(13,14)(15,23)(16,24) (17,20)(18,19)(21,25)(22,26)(27,31)(28,32)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $255$ | $2$ | $( 1, 5)( 2, 6)( 3,30)( 4,29)( 7, 9)( 8,10)(11,34)(12,33)(15,24)(16,23)(17,19) (18,20)(21,26)(22,25)(27,32)(28,31)$ |
Group invariants
| Order: | $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |