Group action invariants
| Degree $n$ : | $15$ | |
| Transitive number $t$ : | $76$ | |
| CHM label : | $[3^{5}:2]A(5)$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,11)(2,7)(4,14)(5,10)(8,13), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (5,10,15), (1,13)(2,14)(3,6)(4,7)(8,11)(9,12) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 6: $S_3$ 60: $A_5$ 120: $A_5\times C_2$ 360: $A_5 \times S_3$ 9720: 15T61 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 5: $A_5$
Low degree siblings
15T76, 30T1029 x 2, 30T1035 x 2, 30T1036 x 2, 45T689 x 2, 45T720Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 5,10,15)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 3, 8,13)( 5,10,15)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 3, 8,13)( 5,15,10)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 1, 6,11)( 3, 8,13)( 5,10,15)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $60$ | $3$ | $( 1, 6,11)( 3, 8,13)( 5,15,10)$ |
| $ 3, 3, 3, 3, 1, 1, 1 $ | $10$ | $3$ | $( 1, 6,11)( 3, 8,13)( 4, 9,14)( 5,10,15)$ |
| $ 3, 3, 3, 3, 1, 1, 1 $ | $40$ | $3$ | $( 1, 6,11)( 3, 8,13)( 4, 9,14)( 5,15,10)$ |
| $ 3, 3, 3, 3, 1, 1, 1 $ | $30$ | $3$ | $( 1, 6,11)( 3,13, 8)( 4, 9,14)( 5,15,10)$ |
| $ 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ |
| $ 3, 3, 3, 3, 3 $ | $10$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$ |
| $ 3, 3, 3, 3, 3 $ | $20$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3,13, 8)( 4, 9,14)( 5,15,10)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $243$ | $2$ | $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $135$ | $2$ | $( 1, 2)( 3, 4)( 6, 7)( 8, 9)(11,12)(13,14)$ |
| $ 3, 2, 2, 2, 2, 2, 2 $ | $270$ | $6$ | $( 1, 2)( 3, 4)( 5,10,15)( 6, 7)( 8, 9)(11,12)(13,14)$ |
| $ 6, 2, 2, 2, 1, 1, 1 $ | $540$ | $6$ | $( 1, 2)( 3, 4, 8, 9,13,14)( 6, 7)(11,12)$ |
| $ 6, 3, 2, 2, 2 $ | $540$ | $6$ | $( 1, 2)( 3, 4, 8, 9,13,14)( 5,10,15)( 6, 7)(11,12)$ |
| $ 6, 3, 2, 2, 2 $ | $540$ | $6$ | $( 1, 2)( 3, 4, 8, 9,13,14)( 5,15,10)( 6, 7)(11,12)$ |
| $ 6, 6, 1, 1, 1 $ | $270$ | $6$ | $( 1, 2, 6, 7,11,12)( 3, 4, 8, 9,13,14)$ |
| $ 6, 6, 3 $ | $270$ | $6$ | $( 1, 2, 6, 7,11,12)( 3, 4, 8, 9,13,14)( 5,10,15)$ |
| $ 6, 6, 3 $ | $270$ | $6$ | $( 1, 2, 6, 7,11,12)( 3, 4, 8, 9,13,14)( 5,15,10)$ |
| $ 6, 6, 1, 1, 1 $ | $270$ | $6$ | $( 1, 2, 6, 7,11,12)( 3, 4,13,14, 8, 9)$ |
| $ 6, 6, 3 $ | $540$ | $6$ | $( 1, 2, 6, 7,11,12)( 3, 4,13,14, 8, 9)( 5,10,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1 $ | $405$ | $2$ | $( 1, 2)( 3, 4)( 6,12)( 7,11)( 8,14)( 9,13)(10,15)$ |
| $ 6, 2, 2, 2, 2, 1 $ | $1620$ | $6$ | $( 1, 2)( 3, 4, 8,14,13, 9)( 6,12)( 7,11)(10,15)$ |
| $ 6, 6, 2, 1 $ | $810$ | $6$ | $( 1, 2, 6,12,11, 7)( 3, 4, 8,14,13, 9)(10,15)$ |
| $ 6, 6, 2, 1 $ | $810$ | $6$ | $( 1, 2, 6,12,11, 7)( 3, 4,13, 9, 8,14)(10,15)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $180$ | $3$ | $( 1, 2, 3)( 6, 7, 8)(11,12,13)$ |
| $ 3, 3, 3, 3, 1, 1, 1 $ | $360$ | $3$ | $( 1, 2, 3)( 5,10,15)( 6, 7, 8)(11,12,13)$ |
| $ 9, 1, 1, 1, 1, 1, 1 $ | $360$ | $9$ | $( 1, 2, 8, 6, 7,13,11,12, 3)$ |
| $ 9, 3, 1, 1, 1 $ | $360$ | $9$ | $( 1, 2, 8, 6, 7,13,11,12, 3)( 5,10,15)$ |
| $ 9, 3, 1, 1, 1 $ | $360$ | $9$ | $( 1, 2, 8, 6, 7,13,11,12, 3)( 5,15,10)$ |
| $ 3, 3, 3, 3, 1, 1, 1 $ | $360$ | $3$ | $( 1, 2, 3)( 4, 9,14)( 6, 7, 8)(11,12,13)$ |
| $ 3, 3, 3, 3, 3 $ | $360$ | $3$ | $( 1, 2, 3)( 4, 9,14)( 5,10,15)( 6, 7, 8)(11,12,13)$ |
| $ 3, 3, 3, 3, 3 $ | $360$ | $3$ | $( 1, 2, 3)( 4, 9,14)( 5,15,10)( 6, 7, 8)(11,12,13)$ |
| $ 9, 3, 1, 1, 1 $ | $360$ | $9$ | $( 1, 2, 8, 6, 7,13,11,12, 3)( 4, 9,14)$ |
| $ 9, 3, 3 $ | $360$ | $9$ | $( 1, 2, 8, 6, 7,13,11,12, 3)( 4, 9,14)( 5,10,15)$ |
| $ 9, 3, 3 $ | $360$ | $9$ | $( 1, 2, 8, 6, 7,13,11,12, 3)( 4, 9,14)( 5,15,10)$ |
| $ 9, 3, 1, 1, 1 $ | $360$ | $9$ | $( 1, 2,13,11,12, 8, 6, 7, 3)( 4, 9,14)$ |
| $ 9, 3, 3 $ | $360$ | $9$ | $( 1, 2,13,11,12, 8, 6, 7, 3)( 4, 9,14)( 5,10,15)$ |
| $ 9, 3, 3 $ | $360$ | $9$ | $( 1, 2,13,11,12, 8, 6, 7, 3)( 4, 9,14)( 5,15,10)$ |
| $ 6, 3, 2, 2, 1, 1 $ | $4860$ | $6$ | $( 1, 2, 3)( 6,12, 8,11, 7,13)( 9,14)(10,15)$ |
| $ 5, 5, 5 $ | $972$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)$ |
| $ 15 $ | $1944$ | $15$ | $( 1, 2, 3, 4,10, 6, 7, 8, 9,15,11,12,13,14, 5)$ |
| $ 10, 5 $ | $2916$ | $10$ | $( 1, 2, 3, 4, 5)( 6,12, 8,14,10,11, 7,13, 9,15)$ |
| $ 5, 5, 5 $ | $972$ | $5$ | $( 1, 2, 3, 5, 4)( 6, 7, 8,10, 9)(11,12,13,15,14)$ |
| $ 15 $ | $1944$ | $15$ | $( 1, 2, 3,10, 9, 6, 7, 8,15,14,11,12,13, 5, 4)$ |
| $ 10, 5 $ | $2916$ | $10$ | $( 1, 2, 3, 5, 4)( 6,12, 8,15, 9,11, 7,13,10,14)$ |
Group invariants
| Order: | $29160=2^{3} \cdot 3^{6} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |