Group action invariants
| Degree $n$ : | $20$ | |
| Transitive number $t$ : | $365$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,12,15,2,11,16)(3,17,19,4,18,20)(5,10,13,6,9,14)(7,8), (1,3,12,10)(2,4,11,9)(5,20,18,14,6,19,17,13)(7,15,8,16), (1,5,8,4)(2,6,7,3)(9,20,10,19)(11,15,14,18)(12,16,13,17) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 120: $S_5$ 240: $S_5\times C_2$ 3840: 20T276 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: None
Degree 10: $S_5$
Low degree siblings
20T365 x 7, 24T9643 x 2, 24T9667 x 2, 40T5418 x 4, 40T5438 x 4, 40T5445 x 4, 40T5469 x 4, 40T5470 x 4, 40T5488 x 8, 40T5550 x 4, 40T5552 x 4Siblings 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 $ | $15$ | $2$ | $( 7, 8)(13,14)(15,16)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $2$ | $( 3, 4)( 9,10)(13,14)(15,16)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)(17,18)(19,20)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $120$ | $4$ | $( 3, 9)( 4,10)( 5,19, 6,20)( 7, 8)(13,18,14,17)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $40$ | $2$ | $( 3, 9)( 4,10)( 5,20)( 6,19)(13,18)(14,17)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $120$ | $4$ | $( 3,10, 4, 9)( 5,19, 6,20)(11,12)(13,17)(14,18)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $40$ | $2$ | $( 1, 2)( 3, 9)( 4,10)( 5,20)( 6,19)( 7, 8)(11,12)(13,17)(14,18)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $40$ | $2$ | $( 3, 9)( 4,10)( 5,20)( 6,19)( 7, 8)(13,17)(14,18)$ |
| $ 4, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $120$ | $4$ | $( 3, 9)( 4,10)( 5,19, 6,20)(13,17,14,18)(15,16)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 1, 1 $ | $120$ | $4$ | $( 3,10, 4, 9)( 5,19, 6,20)( 7, 8)(11,12)(13,18)(14,17)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $40$ | $2$ | $( 1, 2)( 3, 9)( 4,10)( 5,20)( 6,19)(11,12)(13,18)(14,17)(15,16)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1 $ | $320$ | $3$ | $( 3, 9,12)( 4,10,11)( 5,16,20)( 6,15,19)( 7,13,18)( 8,14,17)$ |
| $ 6, 6, 6, 2 $ | $320$ | $6$ | $( 1, 2)( 3, 9,11, 4,10,12)( 5,16,19, 6,15,20)( 7,13,17, 8,14,18)$ |
| $ 6, 3, 3, 3, 3, 1, 1 $ | $320$ | $6$ | $( 3, 9,12)( 4,10,11)( 5,15,20)( 6,16,19)( 7,13,17, 8,14,18)$ |
| $ 6, 6, 3, 3, 2 $ | $320$ | $6$ | $( 1, 2)( 3, 9,11, 4,10,12)( 5,15,19, 6,16,20)( 7,13,18)( 8,14,17)$ |
| $ 6, 6, 6, 1, 1 $ | $320$ | $6$ | $( 3,15, 9,19,12, 6)( 4,16,10,20,11, 5)( 7,17,13, 8,18,14)$ |
| $ 6, 6, 3, 3, 2 $ | $320$ | $6$ | $( 1, 2)( 3,15, 9,20,12, 6)( 4,16,10,19,11, 5)( 7,18,14)( 8,17,13)$ |
| $ 6, 6, 3, 3, 1, 1 $ | $320$ | $6$ | $( 3,16,10,19,12, 6)( 4,15, 9,20,11, 5)( 7,18,14)( 8,17,13)$ |
| $ 6, 6, 6, 2 $ | $320$ | $6$ | $( 1, 2)( 3,16,10,20,12, 6)( 4,15, 9,19,11, 5)( 7,17,13, 8,18,14)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 1, 1 $ | $120$ | $4$ | $( 3,19)( 4,20)( 5,12, 6,11)( 7,18)( 8,17)( 9,15,10,16)(13,14)$ |
| $ 4, 4, 4, 2, 2, 1, 1, 1, 1 $ | $120$ | $4$ | $( 3,20, 4,19)( 5,12, 6,11)( 7,18, 8,17)( 9,16)(10,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $120$ | $2$ | $( 3,19)( 4,20)( 5,12)( 6,11)( 7,17)( 8,18)( 9,16)(10,15)(13,14)$ |
| $ 4, 4, 4, 2, 2, 2, 2 $ | $120$ | $4$ | $( 1, 2)( 3,20, 4,19)( 5,11)( 6,12)( 7,17, 8,18)( 9,15,10,16)(13,14)$ |
| $ 4, 4, 4, 2, 2, 2, 1, 1 $ | $240$ | $4$ | $( 3,20, 4,19)( 5,12, 6,11)( 7,17, 8,18)( 9,16)(10,15)(13,14)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $60$ | $4$ | $( 3,19)( 4,20)( 5,12, 6,11)( 7,17)( 8,18)( 9,15,10,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $60$ | $2$ | $( 3,19)( 4,20)( 5,12)( 6,11)( 7,18)( 8,17)( 9,16)(10,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $60$ | $2$ | $( 1, 2)( 3,19)( 4,20)( 5,11)( 6,12)( 7,18)( 8,17)( 9,16)(10,15)(13,14)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 2 $ | $60$ | $4$ | $( 1, 2)( 3,19)( 4,20)( 5,11, 6,12)( 7,17)( 8,18)( 9,15,10,16)(13,14)$ |
| $ 4, 4, 4, 4, 4 $ | $480$ | $4$ | $( 1,11, 4, 9)( 2,12, 3,10)( 5,15,18, 8)( 6,16,17, 7)(13,20,14,19)$ |
| $ 8, 4, 4, 2, 2 $ | $480$ | $8$ | $( 1,11, 3, 9)( 2,12, 4,10)( 5,15,17, 7, 6,16,18, 8)(13,20)(14,19)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $480$ | $4$ | $( 1,11, 4, 9)( 2,12, 3,10)( 5,16,18, 8)( 6,15,17, 7)(13,19)(14,20)$ |
| $ 8, 4, 4, 4 $ | $480$ | $8$ | $( 1,11, 3, 9)( 2,12, 4,10)( 5,16,17, 7, 6,15,18, 8)(13,19,14,20)$ |
| $ 5, 5, 5, 5 $ | $384$ | $5$ | $( 1,11,17, 8, 6)( 2,12,18, 7, 5)( 3,13,19,10,15)( 4,14,20, 9,16)$ |
| $ 10, 10 $ | $384$ | $10$ | $( 1,11,17, 7, 5, 2,12,18, 8, 6)( 3,14,19,10,16, 4,13,20, 9,15)$ |
| $ 10, 5, 5 $ | $384$ | $10$ | $( 1,11,18, 7, 5, 2,12,17, 8, 6)( 3,13,20, 9,15)( 4,14,19,10,16)$ |
| $ 10, 5, 5 $ | $384$ | $10$ | $( 1,11,18, 8, 6)( 2,12,17, 7, 5)( 3,14,20, 9,16, 4,13,19,10,15)$ |
Group invariants
| Order: | $7680=2^{9} \cdot 3 \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |