Group action invariants
| Degree $n$ : | $25$ | |
| Transitive number $t$ : | $29$ | |
| Group : | $C_5\times A_5$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,23,3,25,5,22,2,24,4,21)(6,12,8,14,10,11,7,13,9,15)(16,17,18,19,20), (1,20,13,6,24)(2,16,14,7,25)(3,17,15,8,21)(4,18,11,9,22)(5,19,12,10,23) | |
| $|\Aut(F/K)|$: | $5$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 5: $C_5$ 60: $A_5$ Resolvents shown for degrees $\leq 47$
Subfields
Low degree siblings
30T69Siblings 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, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $(11,16,21)(12,17,22)(13,18,23)(14,19,24)(15,20,25)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(16,21)(17,22)(18,23)(19,24)(20,25)$ |
| $ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
| $ 15, 5, 5 $ | $20$ | $15$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,17,23,14,20,21,12,18,24,15,16,22,13,19,25)$ |
| $ 10, 10, 5 $ | $15$ | $10$ | $( 1, 2, 3, 4, 5)( 6,12, 8,14,10,11, 7,13, 9,15)(16,22,18,24,20,21,17,23,19,25)$ |
| $ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ |
| $ 15, 5, 5 $ | $20$ | $15$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,18,25,12,19,21,13,20,22,14,16,23,15,17,24)$ |
| $ 10, 10, 5 $ | $15$ | $10$ | $( 1, 3, 5, 2, 4)( 6,13,10,12, 9,11, 8,15, 7,14)(16,23,20,22,19,21,18,25,17,24)$ |
| $ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$ |
| $ 15, 5, 5 $ | $20$ | $15$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,19,22,15,18,21,14,17,25,13,16,24,12,20,23)$ |
| $ 10, 10, 5 $ | $15$ | $10$ | $( 1, 4, 2, 5, 3)( 6,14, 7,15, 8,11, 9,12,10,13)(16,24,17,25,18,21,19,22,20,23)$ |
| $ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$ |
| $ 15, 5, 5 $ | $20$ | $15$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,20,24,13,17,21,15,19,23,12,16,25,14,18,22)$ |
| $ 10, 10, 5 $ | $15$ | $10$ | $( 1, 5, 4, 3, 2)( 6,15, 9,13, 7,11,10,14, 8,12)(16,25,19,23,17,21,20,24,18,22)$ |
| $ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 6,15,19,23)( 2, 7,11,20,24)( 3, 8,12,16,25)( 4, 9,13,17,21) ( 5,10,14,18,22)$ |
| $ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 6,15,24,18)( 2, 7,11,25,19)( 3, 8,12,21,20)( 4, 9,13,22,16) ( 5,10,14,23,17)$ |
| $ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 7,12,17,22)( 2, 8,13,18,23)( 3, 9,14,19,24)( 4,10,15,20,25) ( 5, 6,11,16,21)$ |
| $ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 7,12,22,17)( 2, 8,13,23,18)( 3, 9,14,24,19)( 4,10,15,25,20) ( 5, 6,11,21,16)$ |
| $ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 8,14,20,21)( 2, 9,15,16,22)( 3,10,11,17,23)( 4, 6,12,18,24) ( 5, 7,13,19,25)$ |
| $ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 8,14,25,16)( 2, 9,15,21,17)( 3,10,11,22,18)( 4, 6,12,23,19) ( 5, 7,13,24,20)$ |
| $ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 9,11,18,25)( 2,10,12,19,21)( 3, 6,13,20,22)( 4, 7,14,16,23) ( 5, 8,15,17,24)$ |
| $ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 9,11,23,20)( 2,10,12,24,16)( 3, 6,13,25,17)( 4, 7,14,21,18) ( 5, 8,15,22,19)$ |
| $ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1,10,13,16,24)( 2, 6,14,17,25)( 3, 7,15,18,21)( 4, 8,11,19,22) ( 5, 9,12,20,23)$ |
| $ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1,10,13,21,19)( 2, 6,14,22,20)( 3, 7,15,23,16)( 4, 8,11,24,17) ( 5, 9,12,25,18)$ |
Group invariants
| Order: | $300=2^{2} \cdot 3 \cdot 5^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | [300, 22] |
| Character table: Data not available. |