Group action invariants
| Degree $n$: | $25$ | |
| Transitive number $t$: | $25$ | |
| Group: | $D_{25}:C_5$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,17,5,18,4,19,3,20,2,16)(6,14,9,11,7,13,10,15,8,12)(22,25)(23,24), (1,19,10,24,13,4,17,8,22,11,2,20,6,25,14,5,18,9,23,12,3,16,7,21,15) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $10$: $D_{5}$, $C_{10}$ $50$: $D_5\times C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Low degree siblings
There are no siblings 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, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22)$ |
| $ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)(21,24,22,25,23)$ |
| $ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6, 9, 7,10, 8)(11,12,13,14,15)(16,20,19,18,17)(21,23,25,22,24)$ |
| $ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6,10, 9, 8, 7)(11,14,12,15,13)(16,18,20,17,19)(21,22,23,24,25)$ |
| $ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,21,10,22, 9,23, 8,24, 7,25)(11,16,14,18,12,20,15,17,13,19)$ |
| $ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,22, 8,25,10,23, 7,21, 9,24)(11,18,15,19,14,20,13,16,12,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,23)( 7,22)( 8,21)( 9,25)(10,24)(11,20)(12,19)(13,18)(14,17) (15,16)$ |
| $ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,24, 9,21, 7,23,10,25, 8,22)(11,17,12,16,13,20,14,19,15,18)$ |
| $ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,25, 7,24, 8,23, 9,22,10,21)(11,19,13,17,15,20,12,18,14,16)$ |
| $ 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
| $ 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ |
| $ 25 $ | $10$ | $25$ | $( 1, 6,11,18,22, 2, 7,12,19,23, 3, 8,13,20,24, 4, 9,14,16,25, 5,10,15,17,21)$ |
| $ 25 $ | $10$ | $25$ | $( 1, 6,12,16,23, 2, 7,13,17,24, 3, 8,14,18,25, 4, 9,15,19,21, 5,10,11,20,22)$ |
| $ 25 $ | $10$ | $25$ | $( 1, 6,13,19,24, 2, 7,14,20,25, 3, 8,15,16,21, 4, 9,11,17,22, 5,10,12,18,23)$ |
| $ 25 $ | $10$ | $25$ | $( 1, 6,14,17,25, 2, 7,15,18,21, 3, 8,11,19,22, 4, 9,12,20,23, 5,10,13,16,24)$ |
| $ 25 $ | $10$ | $25$ | $( 1, 6,15,20,21, 2, 7,11,16,22, 3, 8,12,17,23, 4, 9,13,18,24, 5,10,14,19,25)$ |
| $ 25 $ | $10$ | $25$ | $( 1,11,23,10,20, 3,13,25, 7,17, 5,15,22, 9,19, 2,12,24, 6,16, 4,14,21, 8,18)$ |
| $ 25 $ | $10$ | $25$ | $( 1,11,25, 6,17, 3,13,22, 8,19, 5,15,24,10,16, 2,12,21, 7,18, 4,14,23, 9,20)$ |
| $ 25 $ | $10$ | $25$ | $( 1,11,22, 7,19, 3,13,24, 9,16, 5,15,21, 6,18, 2,12,23, 8,20, 4,14,25,10,17)$ |
| $ 25 $ | $10$ | $25$ | $( 1,11,24, 8,16, 3,13,21,10,18, 5,15,23, 7,20, 2,12,25, 9,17, 4,14,22, 6,19)$ |
| $ 25 $ | $10$ | $25$ | $( 1,11,21, 9,18, 3,13,23, 6,20, 5,15,25, 8,17, 2,12,22,10,19, 4,14,24, 7,16)$ |
Group invariants
| Order: | $250=2 \cdot 5^{3}$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [250, 6] |
| Character table: not available. |