Group action invariants
| Degree $n$ : | $20$ | |
| Transitive number $t$ : | $130$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,6,12,17,16)(2,5,11,18,15)(3,8,9,19,13,4,7,10,20,14), (1,19,5,13,11,3,17,8,15,10,2,20,6,14,12,4,18,7,16,9) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 5: $C_5$ 8: $D_{4}$ 10: $C_{10}$ x 3 20: 20T3 40: 20T12 80: $C_2^4 : C_5$ 160: $C_2 \times (C_2^4 : C_5)$ x 3 320: 20T72 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 5: $C_5$
Degree 10: $C_{10}$
Low degree siblings
20T130 x 5, 40T451 x 3, 40T484 x 6, 40T485 x 6, 40T530 x 6, 40T531 x 12, 40T532 x 24, 40T582 x 6, 40T583 x 12, 40T584 x 24Siblings 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 $ | $5$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5, 6)( 7, 8)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1,12,16, 6,17)( 2,11,15, 5,18)( 3, 9,13, 7,20)( 4,10,14, 8,19)$ |
| $ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1,16,17,12, 6)( 2,15,18,11, 5)( 3,13,20, 9, 7)( 4,14,19,10, 8)$ |
| $ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1,17, 6,16,12)( 2,18, 5,15,11)( 3,20, 7,13, 9)( 4,19, 8,14,10)$ |
| $ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 6,12,17,16)( 2, 5,11,18,15)( 3, 7, 9,20,13)( 4, 8,10,19,14)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $( 3, 4)( 7, 8)(11,12)(15,16)(19,20)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $( 3, 4)( 5, 6)(11,12)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(13,14)(19,20)$ |
| $ 10, 5, 5 $ | $32$ | $10$ | $( 1,12,15, 6,17)( 2,11,16, 5,18)( 3,10,14, 8,20, 4, 9,13, 7,19)$ |
| $ 10, 5, 5 $ | $32$ | $10$ | $( 1,16,18,11, 6)( 2,15,17,12, 5)( 3,14,19, 9, 7, 4,13,20,10, 8)$ |
| $ 10, 5, 5 $ | $32$ | $10$ | $( 1,17, 6,16,11)( 2,18, 5,15,12)( 3,19, 7,14,10, 4,20, 8,13, 9)$ |
| $ 10, 5, 5 $ | $32$ | $10$ | $( 1, 6,12,18,15)( 2, 5,11,17,16)( 3, 8, 9,20,14, 4, 7,10,19,13)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 10, 10 $ | $16$ | $10$ | $( 1,12,16, 6,17, 2,11,15, 5,18)( 3, 9,13, 7,20, 4,10,14, 8,19)$ |
| $ 10, 10 $ | $16$ | $10$ | $( 1,16,17,11, 5, 2,15,18,12, 6)( 3,13,20,10, 8, 4,14,19, 9, 7)$ |
| $ 10, 10 $ | $16$ | $10$ | $( 1,17, 5,15,11, 2,18, 6,16,12)( 3,20, 8,14,10, 4,19, 7,13, 9)$ |
| $ 10, 10 $ | $16$ | $10$ | $( 1, 6,12,17,15, 2, 5,11,18,16)( 3, 7, 9,20,14, 4, 8,10,19,13)$ |
| $ 20 $ | $32$ | $20$ | $( 1,19, 5,13,11, 3,17, 8,15,10, 2,20, 6,14,12, 4,18, 7,16, 9)$ |
| $ 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,19,18,20)$ |
| $ 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)$ |
| $ 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,20,18,19)$ |
| $ 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,20,18,19)$ |
| $ 20 $ | $32$ | $20$ | $( 1,10,15, 8,18, 3,12,14, 6,19, 2, 9,16, 7,17, 4,11,13, 5,20)$ |
| $ 20 $ | $32$ | $20$ | $( 1, 8,11,20,16, 3, 6,10,18,13, 2, 7,12,19,15, 4, 5, 9,17,14)$ |
| $ 20 $ | $32$ | $20$ | $( 1,14,17,10, 5, 3,16,20,12, 8, 2,13,18, 9, 6, 4,15,19,11, 7)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1,20, 6,14,11, 4,18, 8,16, 9)( 2,19, 5,13,12, 3,17, 7,15,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1, 9,16, 7,18, 4,11,14, 5,20)( 2,10,15, 8,17, 3,12,13, 6,19)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1, 7,11,19,15, 4, 5, 9,17,13)( 2, 8,12,20,16, 3, 6,10,18,14)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1,13,17, 9, 6, 4,15,19,11, 8)( 2,14,18,10, 5, 3,16,20,12, 7)$ |
Group invariants
| Order: | $640=2^{7} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [640, 21469] |
| Character table: Data not available. |