Group action invariants
| Degree $n$ : | $15$ | |
| Transitive number $t$ : | $43$ | |
| CHM label : | $[3^{4}:2]D(5)$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,6,11)(4,14,9), (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) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 10: $D_{5}$ 20: $D_{10}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 5: $D_{5}$
Low degree siblings
15T43 x 7, 30T290 x 8, 30T291 x 8, 30T297 x 8, 45T205 x 16, 45T206 x 2Siblings 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 $ | $10$ | $3$ | $( 1,11, 6)( 4, 9,14)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 2, 7,12)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 1, 6,11)( 2, 7,12)( 4, 9,14)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 1, 6,11)( 4, 9,14)( 5,10,15)$ |
| $ 3, 3, 3, 3, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 2, 7,12)( 4,14, 9)( 5,10,15)$ |
| $ 3, 3, 3, 3, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 2,12, 7)( 4, 9,14)( 5,10,15)$ |
| $ 3, 3, 3, 3, 1, 1, 1 $ | $10$ | $3$ | $( 1, 6,11)( 2,12, 7)( 4,14, 9)( 5,10,15)$ |
| $ 3, 3, 3, 3, 3 $ | $10$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$ |
| $ 5, 5, 5 $ | $162$ | $5$ | $( 1, 4,12,10, 3)( 2,15, 8, 6, 9)( 5,13,11,14, 7)$ |
| $ 5, 5, 5 $ | $162$ | $5$ | $( 1,12, 3, 4,10)( 2, 8, 9,15, 6)( 5,11, 7,13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $45$ | $2$ | $( 2,15)( 3, 4)( 5, 7)( 8, 9)(10,12)(13,14)$ |
| $ 6, 3, 2, 2, 2 $ | $90$ | $6$ | $( 1,11, 6)( 2,15)( 3, 9, 8,14,13, 4)( 5, 7)(10,12)$ |
| $ 6, 6, 1, 1, 1 $ | $90$ | $6$ | $( 2,15, 7, 5,12,10)( 3,14,13, 9, 8, 4)$ |
| $ 6, 3, 2, 2, 2 $ | $90$ | $6$ | $( 1,11, 6)( 2,15, 7, 5,12,10)( 3, 4)( 8, 9)(13,14)$ |
| $ 6, 6, 3 $ | $90$ | $6$ | $( 1, 6,11)( 2,15, 7, 5,12,10)( 3, 9, 8,14,13, 4)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $81$ | $2$ | $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$ |
| $ 10, 5 $ | $162$ | $10$ | $( 1, 4,12, 5,13, 6,14, 2,15, 3)( 7,10, 8,11, 9)$ |
| $ 10, 5 $ | $162$ | $10$ | $( 1,12,13, 9, 5,11, 2, 8,14,15)( 3, 4,10, 6, 7)$ |
| $ 6, 2, 2, 2, 2, 1 $ | $90$ | $6$ | $( 2,15,12, 5, 7,10)( 3, 4)( 6,11)( 8,14)( 9,13)$ |
| $ 6, 6, 2, 1 $ | $90$ | $6$ | $( 1,11)( 2,15,12, 5, 7,10)( 3, 9,13,14, 8, 4)$ |
| $ 6, 6, 2, 1 $ | $90$ | $6$ | $( 1, 6)( 2,15,12, 5, 7,10)( 3,14, 8, 9,13, 4)$ |
| $ 6, 2, 2, 2, 2, 1 $ | $90$ | $6$ | $( 2,15)( 3,14, 8, 9,13, 4)( 5,12)( 6,11)( 7,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1 $ | $45$ | $2$ | $( 1,11)( 2,15)( 3, 4)( 5,12)( 7,10)( 8,14)( 9,13)$ |
Group invariants
| Order: | $1620=2^{2} \cdot 3^{4} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [1620, 422] |
| Character table: Data not available. |