Group action invariants
| Degree $n$ : | $20$ | |
| Transitive number $t$ : | $531$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,16,19)(2,15,20)(3,18,13)(4,17,14)(5,9,8)(6,10,7), (1,12,10,3)(2,11,9,4)(5,16,8,19,6,15,7,20)(13,18,14,17) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 720: $S_6$ 11520: 16T1753 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: None
Degree 10: $S_{6}$
Low degree siblings
12T285 x 2, 20T531 x 3, 24T12618, 32T1120025 x 4, 40T14226 x 2, 40T14229, 40T14232 x 2, 40T14235, 40T14236 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $(13,14)(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(13,14)(15,16)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $360$ | $4$ | $( 1, 2)( 7, 8)( 9,11,10,12)(13,16,14,15)(17,20)(18,19)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $60$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $60$ | $2$ | $( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)$ |
| $ 4, 4, 4, 2, 2, 2, 1, 1 $ | $720$ | $4$ | $( 3, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,19,14,20)(15,18)(16,17)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $180$ | $4$ | $( 5, 8)( 6, 7)( 9,11)(10,12)(13,19,14,20)(15,17,16,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $180$ | $2$ | $( 5, 7)( 6, 8)( 9,11)(10,12)(13,20)(14,19)(15,17)(16,18)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 2 $ | $180$ | $4$ | $( 1, 2)( 3, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,19,14,20)(15,18,16,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $180$ | $2$ | $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,20)(14,19)(15,18)(16,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $180$ | $2$ | $( 5,10)( 6, 9)( 7,12)( 8,11)(13,14)(15,17)(16,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $30$ | $2$ | $( 5,10)( 6, 9)( 7,12)( 8,11)(15,18)(16,17)$ |
| $ 4, 4, 4, 2, 1, 1, 1, 1, 1, 1 $ | $120$ | $4$ | $( 5,10, 6, 9)( 7,12, 8,11)(13,14)(15,17,16,18)$ |
| $ 4, 4, 4, 2, 2, 2, 1, 1 $ | $120$ | $4$ | $( 3, 4)( 5, 9, 6,10)( 7,12, 8,11)(13,14)(15,18,16,17)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $30$ | $2$ | $( 1, 2)( 3, 4)( 5,10)( 6, 9)( 7,12)( 8,11)(13,14)(15,18)(16,17)(19,20)$ |
| $ 8, 4, 4, 2, 1, 1 $ | $1440$ | $8$ | $( 1, 2)( 5,10, 7,11)( 6, 9, 8,12)(13,18,20,15,14,17,19,16)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $720$ | $4$ | $( 1, 2)( 3, 4)( 5, 9, 7,11)( 6,10, 8,12)(13,17,19,16)(14,18,20,15)$ |
| $ 4, 4, 4, 4, 1, 1, 1, 1 $ | $720$ | $4$ | $( 5, 9, 7,11)( 6,10, 8,12)(13,18,20,16)(14,17,19,15)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1 $ | $640$ | $3$ | $( 3, 6, 7)( 4, 5, 8)( 9,14,18)(10,13,17)(11,15,19)(12,16,20)$ |
| $ 6, 6, 6, 2 $ | $640$ | $6$ | $( 1, 2)( 3, 6, 8, 4, 5, 7)( 9,14,18,10,13,17)(11,16,20,12,15,19)$ |
| $ 6, 6, 6, 2 $ | $1920$ | $6$ | $( 1, 2)( 3, 6, 7, 4, 5, 8)( 9,15,17,12,14,19)(10,16,18,11,13,20)$ |
| $ 6, 6, 3, 3, 1, 1 $ | $1920$ | $6$ | $( 3, 6, 8)( 4, 5, 7)( 9,16,18,11,13,20)(10,15,17,12,14,19)$ |
| $ 6, 6, 3, 3, 1, 1 $ | $960$ | $6$ | $( 3, 6,14,17,19,12)( 4, 5,13,18,20,11)( 7,16, 9)( 8,15,10)$ |
| $ 12, 6, 1, 1 $ | $960$ | $12$ | $( 3, 6,13,17,20,11, 4, 5,14,18,19,12)( 7,15,10, 8,16, 9)$ |
| $ 6, 6, 6, 2 $ | $960$ | $6$ | $( 1, 2)( 3, 6,14,17,19,12)( 4, 5,13,18,20,11)( 7,15, 9, 8,16,10)$ |
| $ 12, 3, 3, 2 $ | $960$ | $12$ | $( 1, 2)( 3, 6,13,17,20,11, 4, 5,14,18,19,12)( 7,16,10)( 8,15, 9)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1 $ | $160$ | $3$ | $( 3,14,19)( 4,13,20)( 5,18,11)( 6,17,12)( 7, 9,16)( 8,10,15)$ |
| $ 6, 6, 3, 3, 1, 1 $ | $480$ | $6$ | $( 3,13,19)( 4,14,20)( 5,17,12, 6,18,11)( 7, 9,15, 8,10,16)$ |
| $ 6, 3, 3, 3, 3, 2 $ | $480$ | $6$ | $( 1, 2)( 3,14,19)( 4,13,20)( 5,18,11)( 6,17,12)( 7,10,16, 8, 9,15)$ |
| $ 6, 6, 6, 2 $ | $160$ | $6$ | $( 1, 2)( 3,14,20, 4,13,19)( 5,18,11, 6,17,12)( 7,10,15, 8, 9,16)$ |
| $ 8, 4, 4, 4 $ | $1440$ | $8$ | $( 1, 4, 2, 3)( 5, 9, 7,12)( 6,10, 8,11)(13,15,20,18,14,16,19,17)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $720$ | $4$ | $( 1, 3)( 2, 4)( 5,10, 7,12)( 6, 9, 8,11)(13,15,20,17)(14,16,19,18)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $720$ | $4$ | $( 1, 3)( 2, 4)( 5,10, 8,11)( 6, 9, 7,12)(13,16,20,17)(14,15,19,18)$ |
| $ 10, 10 $ | $2304$ | $10$ | $( 1, 3, 6,17,10, 2, 4, 5,18, 9)( 7,19,15,14,11, 8,20,16,13,12)$ |
| $ 5, 5, 5, 5 $ | $2304$ | $5$ | $( 1, 3, 6,18, 9)( 2, 4, 5,17,10)( 7,20,15,13,12)( 8,19,16,14,11)$ |
Group invariants
| Order: | $23040=2^{9} \cdot 3^{2} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |