Group action invariants
| Degree $n$ : | $40$ | |
| Transitive number $t$ : | $17$ | |
| Group : | $C_{20}.C_4$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,37,4,40,2,38,3,39)(5,34,7,35,6,33,8,36)(9,30,11,31,10,29,12,32)(13,27,15,25,14,28,16,26)(17,23,20,21,18,24,19,22), (1,16,4,13,2,15,3,14)(5,10,7,12,6,9,8,11)(17,39,20,37,18,40,19,38)(21,34,24,35,22,33,23,36)(25,31,28,29,26,32,27,30) | |
| $|\Aut(F/K)|$: | $20$ |
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: $D_{5}$ 16: $C_8:C_2$ 20: $D_{10}$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $D_{5}$
Degree 8: $C_8:C_2$
Degree 10: $D_5$
Degree 20: 20T2
Low degree siblings
There are no siblings with degree $\leq 10$
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, 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, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 5, 6)( 7, 8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)(37,38)(39,40)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,24,22,23) (25,27,26,28)(29,32,30,31)(33,35,34,36)(37,40,38,39)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,31,30,32)(33,35,34,36)(37,39,38,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)$ |
| $ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1, 5, 3, 8, 2, 6, 4, 7)( 9,38,12,40,10,37,11,39)(13,34,16,36,14,33,15,35) (17,30,19,32,18,29,20,31)(21,25,23,27,22,26,24,28)$ |
| $ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1, 5, 4, 7, 2, 6, 3, 8)( 9,38,11,39,10,37,12,40)(13,33,15,36,14,34,16,35) (17,30,20,31,18,29,19,32)(21,26,24,27,22,25,23,28)$ |
| $ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1, 7, 4, 6, 2, 8, 3, 5)( 9,39,11,37,10,40,12,38)(13,35,15,33,14,36,16,34) (17,31,20,29,18,32,19,30)(21,28,24,26,22,27,23,25)$ |
| $ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1, 7, 3, 5, 2, 8, 4, 6)( 9,39,12,38,10,40,11,37)(13,36,16,33,14,35,15,34) (17,31,19,30,18,32,20,29)(21,27,23,26,22,28,24,25)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 9,18,25,35)( 2,10,17,26,36)( 3,12,20,27,34)( 4,11,19,28,33) ( 5,13,23,29,38)( 6,14,24,30,37)( 7,15,21,32,39)( 8,16,22,31,40)$ |
| $ 10, 10, 5, 5, 5, 5 $ | $2$ | $10$ | $( 1, 9,18,25,35)( 2,10,17,26,36)( 3,12,20,27,34)( 4,11,19,28,33) ( 5,14,23,30,38, 6,13,24,29,37)( 7,16,21,31,39, 8,15,22,32,40)$ |
| $ 10, 10, 5, 5, 5, 5 $ | $2$ | $10$ | $( 1,10,18,26,35, 2, 9,17,25,36)( 3,11,20,28,34, 4,12,19,27,33)( 5,13,23,29,38) ( 6,14,24,30,37)( 7,15,21,32,39)( 8,16,22,31,40)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,10,18,26,35, 2, 9,17,25,36)( 3,11,20,28,34, 4,12,19,27,33)( 5,14,23,30,38, 6,13,24,29,37)( 7,16,21,31,39, 8,15,22,32,40)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,11,17,27,35, 4,10,20,25,33, 2,12,18,28,36, 3, 9,19,26,34)( 5,15,24,31,38, 7,14,22,29,39, 6,16,23,32,37, 8,13,21,30,40)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,11,17,27,35, 4,10,20,25,33, 2,12,18,28,36, 3, 9,19,26,34)( 5,16,24,32,38, 8,14,21,29,40, 6,15,23,31,37, 7,13,22,30,39)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,12,17,28,35, 3,10,19,25,34, 2,11,18,27,36, 4, 9,20,26,33)( 5,15,24,31,38, 7,14,22,29,39, 6,16,23,32,37, 8,13,21,30,40)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,12,17,28,35, 3,10,19,25,34, 2,11,18,27,36, 4, 9,20,26,33)( 5,16,24,32,38, 8,14,21,29,40, 6,15,23,31,37, 7,13,22,30,39)$ |
| $ 10, 10, 5, 5, 5, 5 $ | $2$ | $10$ | $( 1,17,35,10,25, 2,18,36, 9,26)( 3,19,34,11,27, 4,20,33,12,28)( 5,23,38,13,29) ( 6,24,37,14,30)( 7,21,39,15,32)( 8,22,40,16,31)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,17,35,10,25, 2,18,36, 9,26)( 3,19,34,11,27, 4,20,33,12,28)( 5,24,38,14,29, 6,23,37,13,30)( 7,22,39,16,32, 8,21,40,15,31)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,18,35, 9,25)( 2,17,36,10,26)( 3,20,34,12,27)( 4,19,33,11,28) ( 5,23,38,13,29)( 6,24,37,14,30)( 7,21,39,15,32)( 8,22,40,16,31)$ |
| $ 10, 10, 5, 5, 5, 5 $ | $2$ | $10$ | $( 1,18,35, 9,25)( 2,17,36,10,26)( 3,20,34,12,27)( 4,19,33,11,28) ( 5,24,38,14,29, 6,23,37,13,30)( 7,22,39,16,32, 8,21,40,15,31)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,19,36,12,25, 4,17,34, 9,28, 2,20,35,11,26, 3,18,33,10,27)( 5,21,37,16,29, 7,24,40,13,32, 6,22,38,15,30, 8,23,39,14,31)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,19,36,12,25, 4,17,34, 9,28, 2,20,35,11,26, 3,18,33,10,27)( 5,22,37,15,29, 8,24,39,13,31, 6,21,38,16,30, 7,23,40,14,32)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,20,36,11,25, 3,17,33, 9,27, 2,19,35,12,26, 4,18,34,10,28)( 5,21,37,16,29, 7,24,40,13,32, 6,22,38,15,30, 8,23,39,14,31)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,20,36,11,25, 3,17,33, 9,27, 2,19,35,12,26, 4,18,34,10,28)( 5,22,37,15,29, 8,24,39,13,31, 6,21,38,16,30, 7,23,40,14,32)$ |
Group invariants
| Order: | $80=2^{4} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [80, 10] |
| Character table: Data not available. |