Group action invariants
| Degree $n$ : | $20$ | |
| Transitive number $t$ : | $141$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,15,19,14,8,12,5,9,4,17)(2,16,20,13,7,11,6,10,3,18), (1,6,19,14,8,2,5,20,13,7)(3,18,12,16,9,4,17,11,15,10), (1,13)(2,14)(3,12)(4,11)(5,19,16,10)(6,20,15,9) | |
| $|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_2^2$ x 7 8: $C_2^3$ 10: $D_{5}$ 20: $D_{10}$ x 3 40: 20T8 160: $(C_2^4 : C_5) : C_2$ 320: $C_2\times (C_2^4 : D_5)$ x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 5: $D_{5}$
Degree 10: $D_{10}$, $C_2\times (C_2^4 : D_5)$ x 2
Low degree siblings
20T141 x 35, 20T143 x 24, 40T458 x 9, 40T470 x 12, 40T471 x 12, 40T540 x 36, 40T549 x 36, 40T550 x 36, 40T571 x 18, 40T575 x 6, 40T580 x 36Siblings 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, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 1,11)( 2,12)( 5,16)( 6,15)( 7,17)( 8,18)( 9,20)(10,19)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 1,11)( 2,12)( 3,14)( 4,13)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 1,11)( 2,12)( 5,16)( 6,15)$ |
| $ 5, 5, 5, 5 $ | $32$ | $5$ | $( 1,19, 8, 5, 4)( 2,20, 7, 6, 3)( 9,17,15,14,12)(10,18,16,13,11)$ |
| $ 5, 5, 5, 5 $ | $32$ | $5$ | $( 1, 8, 4,19, 5)( 2, 7, 3,20, 6)( 9,15,12,17,14)(10,16,11,18,13)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1,15,19,14, 8,12, 5, 9, 4,17)( 2,16,20,13, 7,11, 6,10, 3,18)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1,14, 5,17,19,12, 4,15, 8, 9)( 2,13, 6,18,20,11, 3,16, 7,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,18)( 8,17)( 9,19)(10,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7, 8)( 9,10)(11,12)(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7,18)( 8,17)( 9,19)(10,20)(11,12)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,18)( 8,17)( 9,19)(10,20)(11,12)(15,16)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 9,20)(10,19)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 1,11)( 2,12)( 5,16)( 6,15)( 7,17)( 8,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,20)(10,19)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1,19,18,16,13,11,10, 8, 5, 4)( 2,20,17,15,14,12, 9, 7, 6, 3)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1, 8, 4,19,16,11,18,13,10, 5)( 2, 7, 3,20,15,12,17,14, 9, 6)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1,15,19, 3,18, 2,16,20, 4,17)( 5, 9,13, 7,11, 6,10,14, 8,12)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1,14, 5,17,19, 2,13, 6,18,20)( 3,16, 7,10,12, 4,15, 8, 9,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,18)( 8,17)( 9,10)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7, 8)( 9,19)(10,20)(11,12)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1,12)( 2,11)( 3, 4)( 5, 6)( 7, 8)( 9,19)(10,20)(13,14)(15,16)(17,18)$ |
| $ 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 $ | $20$ | $4$ | $( 3,20,14, 9)( 4,19,13,10)( 5,18)( 6,17)( 7,15)( 8,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $2$ | $( 1,11)( 2,12)( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,19)(14,20)(15,17)(16,18)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $20$ | $4$ | $( 3, 9)( 4,10)( 5, 8,16,18)( 6, 7,15,17)(13,19)(14,20)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $20$ | $4$ | $( 1,11)( 2,12)( 3,20,14, 9)( 4,19,13,10)( 5,18,16, 8)( 6,17,15, 7)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1,15,11, 6)( 2,16,12, 5)( 3,13)( 4,14)( 7,19)( 8,20)( 9,18)(10,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9)(11,15)(12,16)(13,14)(17,19)(18,20)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1, 6)( 2, 5)( 3,13)( 4,14)( 7,10,17,19)( 8, 9,18,20)(11,15)(12,16)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $20$ | $4$ | $( 1,15,11, 6)( 2,16,12, 5)( 3, 4)( 7,19,17,10)( 8,20,18, 9)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $20$ | $2$ | $( 3, 9)( 4,10)( 5,18)( 6,17)( 7,15)( 8,16)(13,19)(14,20)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1,11)( 2,12)( 3,20,14, 9)( 4,19,13,10)( 5, 8)( 6, 7)(15,17)(16,18)$ |
| $ 4, 4, 4, 4, 1, 1, 1, 1 $ | $20$ | $4$ | $( 3,20,14, 9)( 4,19,13,10)( 5, 8,16,18)( 6, 7,15,17)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,18,16, 8)( 6,17,15, 7)(13,19)(14,20)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1,15,11, 6)( 2,16,12, 5)( 3, 4)( 7,19)( 8,20)( 9,18)(10,17)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $2$ | $( 1, 6)( 2, 5)( 3,13)( 4,14)( 7,10)( 8, 9)(11,15)(12,16)(17,19)(18,20)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1, 6)( 2, 5)( 3, 4)( 7,10,17,19)( 8, 9,18,20)(11,15)(12,16)(13,14)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $20$ | $4$ | $( 1,15,11, 6)( 2,16,12, 5)( 3,13)( 4,14)( 7,19,17,10)( 8,20,18, 9)$ |
Group invariants
| Order: | $640=2^{7} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [640, 21537] |
| Character table: Data not available. |