Group action invariants
| Degree $n$ : | $20$ | |
| Transitive number $t$ : | $123$ | |
| Group : | $C_4\times S_5$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,15,18,3)(2,16,17,4)(5,8,13,11)(6,7,14,12)(9,19,10,20), (1,4,10,8,6,11,14,20,17,15,2,3,9,7,5,12,13,19,18,16) | |
| $|\Aut(F/K)|$: | $4$ |
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: $C_4\times C_2$ 120: $S_5$ 240: $S_5\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $S_5$
Degree 10: $S_5\times C_2$
Low degree siblings
20T123, 24T1347 x 2, 24T1348 x 2, 40T408 x 2, 40T409 x 2, 40T430Siblings 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 $ | $10$ | $2$ | $( 7,19)( 8,20)( 9,17)(10,18)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 5,10,18)( 6, 9,17)( 7,15,19)( 8,16,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $15$ | $2$ | $( 3, 8)( 4, 7)( 5,10)( 6, 9)(13,17)(14,18)(15,19)(16,20)$ |
| $ 4, 4, 4, 4, 1, 1, 1, 1 $ | $30$ | $4$ | $( 3, 8,16,20)( 4, 7,15,19)( 5,10,14,18)( 6, 9,13,17)$ |
| $ 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, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,20)( 8,19)( 9,18)(10,17)(11,12)(13,14)(15,16)$ |
| $ 6, 6, 2, 2, 2, 2 $ | $20$ | $6$ | $( 1, 2)( 3, 4)( 5, 9,18, 6,10,17)( 7,16,19, 8,15,20)(11,12)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3, 7)( 4, 8)( 5, 9)( 6,10)(11,12)(13,18)(14,17)(15,20)(16,19)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $30$ | $4$ | $( 1, 2)( 3, 7,16,19)( 4, 8,15,20)( 5, 9,14,17)( 6,10,13,18)(11,12)$ |
| $ 4, 4, 4, 4, 4 $ | $15$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,20,10,19)(11,13,12,14)(15,17,16,18)$ |
| $ 12, 4, 4 $ | $20$ | $12$ | $( 1, 3, 2, 4)( 5, 7, 9,16,18,19, 6, 8,10,15,17,20)(11,13,12,14)$ |
| $ 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 3, 2, 4)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)(11,13,12,14)$ |
| $ 4, 4, 4, 4, 4 $ | $30$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,20,10,19)(11,13,16,18)(12,14,15,17)$ |
| $ 20 $ | $24$ | $20$ | $( 1, 3, 5, 7, 9,12,14,15,17,20, 2, 4, 6, 8,10,11,13,16,18,19)$ |
| $ 12, 4, 4 $ | $20$ | $12$ | $( 1, 3, 5,11,13,16, 2, 4, 6,12,14,15)( 7,17, 8,18)( 9,20,10,19)$ |
| $ 4, 4, 4, 4, 4 $ | $15$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,19,10,20)(11,14,12,13)(15,18,16,17)$ |
| $ 12, 4, 4 $ | $20$ | $12$ | $( 1, 4, 2, 3)( 5, 8, 9,15,18,20, 6, 7,10,16,17,19)(11,14,12,13)$ |
| $ 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 4, 2, 3)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)(11,14,12,13)$ |
| $ 4, 4, 4, 4, 4 $ | $30$ | $4$ | $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,19,10,20)(11,14,16,17)(12,13,15,18)$ |
| $ 20 $ | $24$ | $20$ | $( 1, 4, 5, 8, 9,11,14,16,17,19, 2, 3, 6, 7,10,12,13,15,18,20)$ |
| $ 12, 4, 4 $ | $20$ | $12$ | $( 1, 4, 5,12,13,15, 2, 3, 6,11,14,16)( 7,18, 8,17)( 9,19,10,20)$ |
| $ 6, 6, 2, 2, 2, 2 $ | $20$ | $6$ | $( 1, 5)( 2, 6)( 3, 7,20, 4, 8,19)( 9,14,17,10,13,18)(11,16)(12,15)$ |
| $ 10, 10 $ | $24$ | $10$ | $( 1, 5, 9,14,17, 2, 6,10,13,18)( 3, 7,12,15,20, 4, 8,11,16,19)$ |
| $ 3, 3, 3, 3, 2, 2, 2, 2 $ | $20$ | $6$ | $( 1, 6)( 2, 5)( 3, 8,20)( 4, 7,19)( 9,13,17)(10,14,18)(11,15)(12,16)$ |
| $ 5, 5, 5, 5 $ | $24$ | $5$ | $( 1, 6, 9,13,17)( 2, 5,10,14,18)( 3, 8,12,16,20)( 4, 7,11,15,19)$ |
| $ 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,11, 2,12)( 3,13, 4,14)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)$ |
| $ 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,12, 2,11)( 3,14, 4,13)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$ |
Group invariants
| Order: | $480=2^{5} \cdot 3 \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | [480, 943] |
| Character table: Data not available. |