Group action invariants
| Degree $n$ : | $15$ | |
| Transitive number $t$ : | $53$ | |
| CHM label : | $[3^{4}]A(5)$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,6,11)(4,14,9), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (1,13)(2,14)(3,6)(4,7)(8,11)(9,12) | |
| $|\Aut(F/K)|$: | $3$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 60: $A_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 5: $A_5$
Low degree siblings
15T53 x 2, 30T558, 30T562 x 3, 45T363, 45T364Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 4, 9,14)( 5,15,10)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 2, 7,12)( 4, 9,14)( 5,10,15)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 2,12, 7)( 4,14, 9)( 5,15,10)$ |
| $ 3, 3, 3, 3, 1, 1, 1 $ | $30$ | $3$ | $( 2, 7,12)( 3,13, 8)( 4, 9,14)( 5,15,10)$ |
| $ 3, 3, 3, 3, 3 $ | $5$ | $3$ | $( 1,11, 6)( 2, 7,12)( 3,13, 8)( 4,14, 9)( 5,15,10)$ |
| $ 3, 3, 3, 3, 3 $ | $5$ | $3$ | $( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $135$ | $2$ | $( 1, 2)( 3, 4)( 6, 7)( 8, 9)(11,12)(13,14)$ |
| $ 6, 3, 2, 2, 2 $ | $270$ | $6$ | $( 1, 2)( 3, 9, 8,14,13, 4)( 5,15,10)( 6, 7)(11,12)$ |
| $ 6, 3, 2, 2, 2 $ | $270$ | $6$ | $( 1, 2)( 3,14,13, 9, 8, 4)( 5,10,15)( 6, 7)(11,12)$ |
| $ 6, 6, 3 $ | $135$ | $6$ | $( 1, 7, 6,12,11, 2)( 3, 9, 8,14,13, 4)( 5,10,15)$ |
| $ 6, 6, 1, 1, 1 $ | $270$ | $6$ | $( 1, 7, 6,12,11, 2)( 3,14,13, 9, 8, 4)$ |
| $ 6, 6, 3 $ | $135$ | $6$ | $( 1,12,11, 7, 6, 2)( 3,14,13, 9, 8, 4)( 5,15,10)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $180$ | $3$ | $( 1, 2, 3)( 6, 7, 8)(11,12,13)$ |
| $ 3, 3, 3, 3, 3 $ | $180$ | $3$ | $( 1, 2, 3)( 4, 9,14)( 5,15,10)( 6, 7, 8)(11,12,13)$ |
| $ 3, 3, 3, 3, 3 $ | $180$ | $3$ | $( 1, 2, 3)( 4,14, 9)( 5,10,15)( 6, 7, 8)(11,12,13)$ |
| $ 9, 3, 1, 1, 1 $ | $180$ | $9$ | $( 1, 7, 8, 6,12,13,11, 2, 3)( 5,15,10)$ |
| $ 9, 3, 3 $ | $180$ | $9$ | $( 1, 7, 8, 6,12,13,11, 2, 3)( 4, 9,14)( 5,10,15)$ |
| $ 9, 3, 1, 1, 1 $ | $180$ | $9$ | $( 1, 7, 8, 6,12,13,11, 2, 3)( 4,14, 9)$ |
| $ 9, 3, 1, 1, 1 $ | $180$ | $9$ | $( 1,12,13,11, 7, 8, 6, 2, 3)( 5,10,15)$ |
| $ 9, 3, 1, 1, 1 $ | $180$ | $9$ | $( 1,12,13,11, 7, 8, 6, 2, 3)( 4, 9,14)$ |
| $ 9, 3, 3 $ | $180$ | $9$ | $( 1,12,13,11, 7, 8, 6, 2, 3)( 4,14, 9)( 5,15,10)$ |
| $ 5, 5, 5 $ | $972$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)$ |
| $ 5, 5, 5 $ | $972$ | $5$ | $( 1, 2, 3, 5, 4)( 6, 7, 8,10, 9)(11,12,13,15,14)$ |
Group invariants
| Order: | $4860=2^{2} \cdot 3^{5} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |