Group action invariants
| Degree $n$ : | $15$ | |
| Transitive number $t$ : | $38$ | |
| CHM label : | $[5^{3}:4]3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (1,7,4,13)(2,14,8,11)(3,6,12,9), (3,6,9,12,15) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 4: $C_4$ 6: $C_6$ 12: $C_{12}$ 20: $F_5$ 60: $F_5\times C_3$ 300: $(C_5^2 : C_4):C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: None
Low degree siblings
15T38 x 7, 30T287 x 8Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 5, 5, 5 $ | $12$ | $5$ | $( 1,13,10, 7, 4)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$ |
| $ 5, 5, 1, 1, 1, 1, 1 $ | $12$ | $5$ | $( 2,11, 5,14, 8)( 3, 9,15, 6,12)$ |
| $ 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ |
| $ 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$ |
| $ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $5$ | $( 3, 6, 9,12,15)$ |
| $ 5, 5, 1, 1, 1, 1, 1 $ | $12$ | $5$ | $( 1,13,10, 7, 4)( 2, 8,14, 5,11)$ |
| $ 5, 5, 5 $ | $12$ | $5$ | $( 1,10, 4,13, 7)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ |
| $ 5, 5, 5 $ | $12$ | $5$ | $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3, 9,15, 6,12)$ |
| $ 5, 5, 1, 1, 1, 1, 1 $ | $12$ | $5$ | $( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ |
| $ 5, 5, 1, 1, 1, 1, 1 $ | $12$ | $5$ | $( 1,13,10, 7, 4)( 3, 9,15, 6,12)$ |
| $ 5, 5, 5 $ | $12$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$ |
| $ 5, 5, 5 $ | $4$ | $5$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
| $ 5, 5, 5 $ | $12$ | $5$ | $( 1,10, 4,13, 7)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$ |
| $ 15 $ | $100$ | $15$ | $( 1, 6,14, 4, 9, 2, 7,12, 5,10,15, 8,13, 3,11)$ |
| $ 15 $ | $100$ | $15$ | $( 1,11, 6, 4,14, 9, 7, 2,12,10, 5,15,13, 8, 3)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $125$ | $2$ | $( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)$ |
| $ 6, 6, 3 $ | $125$ | $6$ | $( 1, 6, 5, 4, 3, 8)( 2, 7,15,11,13, 9)(10,12,14)$ |
| $ 6, 6, 3 $ | $125$ | $6$ | $( 1,11, 3,13,14,15)( 2,12, 4, 8, 6,10)( 5, 9, 7)$ |
| $ 4, 4, 4, 1, 1, 1 $ | $125$ | $4$ | $( 4, 7,13,10)( 5, 8,14,11)( 6, 9,15,12)$ |
| $ 12, 3 $ | $125$ | $12$ | $( 1, 6,14)( 2, 7, 3, 8, 4,12,11,10, 9, 5,13,15)$ |
| $ 12, 3 $ | $125$ | $12$ | $( 1,11,15, 7, 8, 9,10,14, 6, 4, 2,12)( 3,13, 5)$ |
| $ 4, 4, 4, 1, 1, 1 $ | $125$ | $4$ | $( 4,10,13, 7)( 5,11,14, 8)( 6,12,15, 9)$ |
| $ 12, 3 $ | $125$ | $12$ | $( 1, 6, 2, 7, 9,11, 4,15,14,13,12, 5)( 3, 8,10)$ |
| $ 12, 3 $ | $125$ | $12$ | $( 1,11, 9)( 2,12,10, 8,15, 4, 5, 6, 7,14, 3,13)$ |
Group invariants
| Order: | $1500=2^{2} \cdot 3 \cdot 5^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [1500, 119] |
| Character table: Data not available. |