Group action invariants
| Degree $n$ : | $20$ | |
| Transitive number $t$ : | $144$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,20,12,15,7,4,18,9,13,6,2,19,11,16,8,3,17,10,14,5), (1,5,2,6)(3,8,4,7)(9,12,10,11)(13,20)(14,19)(15,17)(16,18) | |
| $|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_4$ x 2, $C_2^2$ 8: $C_4\times C_2$ 10: $D_{5}$ 20: $D_{10}$ 40: 20T6 160: $(C_2^4 : C_5) : C_2$ 320: $C_2\times (C_2^4 : D_5)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 5: $D_{5}$
Degree 10: $D_{10}$
Low degree siblings
20T144 x 5, 40T455 x 3, 40T464 x 6, 40T465 x 6, 40T533 x 6, 40T535 x 6, 40T544 x 6, 40T545 x 6Siblings 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 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)$ |
| $ 5, 5, 5, 5 $ | $32$ | $5$ | $( 1, 7,13,11,17)( 2, 8,14,12,18)( 3, 5,15, 9,19)( 4, 6,16,10,20)$ |
| $ 5, 5, 5, 5 $ | $32$ | $5$ | $( 1,13,17, 7,11)( 2,14,18, 8,12)( 3,15,19, 5, 9)( 4,16,20, 6,10)$ |
| $ 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 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 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 $ | $32$ | $10$ | $( 1, 7,13,11,17, 2, 8,14,12,18)( 3, 5,15, 9,19, 4, 6,16,10,20)$ |
| $ 10, 10 $ | $32$ | $10$ | $( 1,13,17, 8,12, 2,14,18, 7,11)( 3,15,19, 6,10, 4,16,20, 5, 9)$ |
| $ 20 $ | $32$ | $20$ | $( 1,20,12,15, 7, 4,18, 9,13, 6, 2,19,11,16, 8, 3,17,10,14, 5)$ |
| $ 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,20,18,19)$ |
| $ 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)(17,20,18,19)$ |
| $ 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,19,18,20)$ |
| $ 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)(17,20,18,19)$ |
| $ 20 $ | $32$ | $20$ | $( 1,10, 8,19,13, 4,12, 5,17,16, 2, 9, 7,20,14, 3,11, 6,18,15)$ |
| $ 20 $ | $32$ | $20$ | $( 1,19,11,16, 8, 3,18, 9,13, 6, 2,20,12,15, 7, 4,17,10,14, 5)$ |
| $ 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,20,18,19)$ |
| $ 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,19,18,20)$ |
| $ 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,19,18,20)$ |
| $ 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,19,18,20)$ |
| $ 20 $ | $32$ | $20$ | $( 1, 9, 7,20,14, 3,12, 5,17,16, 2,10, 8,19,13, 4,11, 6,18,15)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $20$ | $4$ | $( 5,19)( 6,20)( 7,17)( 8,18)( 9,15,10,16)(11,13,12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $2$ | $( 1, 2)( 3, 4)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $20$ | $4$ | $( 5,20, 6,19)( 7,18, 8,17)( 9,16)(10,15)(11,14)(12,13)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $20$ | $4$ | $( 1, 2)( 3, 4)( 5,20, 6,19)( 7,18, 8,17)( 9,16,10,15)(11,14,12,13)$ |
| $ 4, 4, 4, 4, 1, 1, 1, 1 $ | $20$ | $4$ | $( 5,20, 6,19)( 7,18, 8,17)( 9,15,10,16)(11,13,12,14)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1, 2)( 3, 4)( 5,20, 6,19)( 7,18, 8,17)( 9,15)(10,16)(11,13)(12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $20$ | $2$ | $( 5,19)( 6,20)( 7,17)( 8,18)( 9,16)(10,15)(11,14)(12,13)$ |
| $ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1, 2)( 3, 4)( 5,19)( 6,20)( 7,17)( 8,18)( 9,16,10,15)(11,14,12,13)$ |
| $ 4, 4, 4, 4, 4 $ | $20$ | $4$ | $( 1,20, 2,19)( 3,17, 4,18)( 5,11, 6,12)( 7,10, 8, 9)(13,15,14,16)$ |
| $ 4, 2, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1,20)( 2,19)( 3,17)( 4,18)( 5,12)( 6,11)( 7, 9)( 8,10)(13,15,14,16)$ |
| $ 4, 4, 4, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1,19)( 2,20)( 3,18)( 4,17)( 5,11, 6,12)( 7,10, 8, 9)(13,16,14,15)$ |
| $ 4, 4, 4, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1,19, 2,20)( 3,18, 4,17)( 5,12)( 6,11)( 7, 9)( 8,10)(13,16,14,15)$ |
| $ 4, 4, 4, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1,19)( 2,20)( 3,18)( 4,17)( 5,11, 6,12)( 7,10, 8, 9)(13,15,14,16)$ |
| $ 4, 4, 4, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1,19, 2,20)( 3,18, 4,17)( 5,12)( 6,11)( 7, 9)( 8,10)(13,15,14,16)$ |
| $ 4, 4, 4, 4, 4 $ | $20$ | $4$ | $( 1,20, 2,19)( 3,17, 4,18)( 5,11, 6,12)( 7,10, 8, 9)(13,16,14,15)$ |
| $ 4, 2, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1,20)( 2,19)( 3,17)( 4,18)( 5,12)( 6,11)( 7, 9)( 8,10)(13,16,14,15)$ |
Group invariants
| Order: | $640=2^{7} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [640, 21458] |
| Character table: Data not available. |