Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $888$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,2)(3,12,5,7,18,4,11,6,8,17)(9,13,16,10,14,15), (1,10,5,13,11,17,16,2,9,6,14,12,18,15)(3,4)(7,8) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 181440: $A_9$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: None
Degree 9: $A_9$
Low degree siblings
There are no siblings 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $378$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $378$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $945$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $945$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $168$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $168$ | $3$ | $( 1, 3, 5)( 2, 4, 6)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $7560$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7, 9)( 8,10)(11,13)(12,14)(15,16)(17,18)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $7560$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7,10)( 8, 9)(11,14)(12,13)$ |
| $ 6, 6, 2, 2, 2 $ | $3360$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7, 9,12, 8,10,11)(13,14)(15,16)(17,18)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3360$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)$ |
| $ 6, 6, 6 $ | $2240$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7, 9,12, 8,10,11)(13,16,17,14,15,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $2240$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,15,17)(14,16,18)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $7560$ | $4$ | $( 1, 4, 5, 7)( 2, 3, 6, 8)( 9,12)(10,11)(13,14)(15,16)(17,18)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $7560$ | $4$ | $( 1, 3, 5, 8)( 2, 4, 6, 7)( 9,11)(10,12)$ |
| $ 6, 4, 4, 2, 2 $ | $15120$ | $12$ | $( 1, 4, 5, 7)( 2, 3, 6, 8)( 9,12,14,10,11,13)(15,18)(16,17)$ |
| $ 4, 4, 3, 3, 2, 2 $ | $15120$ | $12$ | $( 1, 3, 5, 8)( 2, 4, 6, 7)( 9,11,14)(10,12,13)(15,17)(16,18)$ |
| $ 4, 4, 4, 4, 2 $ | $11340$ | $4$ | $( 1, 4, 5, 7)( 2, 3, 6, 8)( 9,12,14,15)(10,11,13,16)(17,18)$ |
| $ 4, 4, 4, 4, 1, 1 $ | $11340$ | $4$ | $( 1, 3, 5, 8)( 2, 4, 6, 7)( 9,11,14,16)(10,12,13,15)$ |
| $ 10, 2, 2, 2, 2 $ | $3024$ | $10$ | $( 1, 4, 5, 7, 9, 2, 3, 6, 8,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3024$ | $5$ | $( 1, 3, 5, 8, 9)( 2, 4, 6, 7,10)$ |
| $ 10, 2, 2, 2, 2 $ | $9072$ | $10$ | $( 1, 4, 5, 7, 9, 2, 3, 6, 8,10)(11,13)(12,14)(15,18)(16,17)$ |
| $ 5, 5, 2, 2, 2, 2 $ | $9072$ | $10$ | $( 1, 3, 5, 8, 9)( 2, 4, 6, 7,10)(11,14)(12,13)(15,17)(16,18)$ |
| $ 10, 6, 2 $ | $12096$ | $30$ | $( 1, 4, 5, 7, 9, 2, 3, 6, 8,10)(11,13,16,12,14,15)(17,18)$ |
| $ 5, 5, 3, 3, 1, 1 $ | $12096$ | $15$ | $( 1, 3, 5, 8, 9)( 2, 4, 6, 7,10)(11,14,16)(12,13,15)$ |
| $ 10, 6, 2 $ | $12096$ | $30$ | $( 1, 4, 5, 7, 9, 2, 3, 6, 8,10)(11,13,18,12,14,17)(15,16)$ |
| $ 5, 5, 3, 3, 1, 1 $ | $12096$ | $15$ | $( 1, 3, 5, 8, 9)( 2, 4, 6, 7,10)(11,14,18)(12,13,17)$ |
| $ 6, 6, 2, 2, 2 $ | $30240$ | $6$ | $( 1, 4, 5, 7, 9,12)( 2, 3, 6, 8,10,11)(13,16)(14,15)(17,18)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $30240$ | $6$ | $( 1, 3, 5, 8, 9,11)( 2, 4, 6, 7,10,12)(13,15)(14,16)$ |
| $ 14, 2, 2 $ | $25920$ | $14$ | $( 1, 4, 5, 7, 9,12,14, 2, 3, 6, 8,10,11,13)(15,16)(17,18)$ |
| $ 7, 7, 1, 1, 1, 1 $ | $25920$ | $7$ | $( 1, 3, 5, 8, 9,11,14)( 2, 4, 6, 7,10,12,13)$ |
| $ 18 $ | $20160$ | $18$ | $( 1, 4, 5, 7, 9,12,14,15,18, 2, 3, 6, 8,10,11,13,16,17)$ |
| $ 9, 9 $ | $20160$ | $9$ | $( 1, 3, 5, 8, 9,11,14,16,18)( 2, 4, 6, 7,10,12,13,15,17)$ |
| $ 18 $ | $20160$ | $18$ | $( 1, 4, 5, 7, 9,12,14,17,16, 2, 3, 6, 8,10,11,13,18,15)$ |
| $ 9, 9 $ | $20160$ | $9$ | $( 1, 3, 5, 8, 9,11,14,18,16)( 2, 4, 6, 7,10,12,13,17,15)$ |
Group invariants
| Order: | $362880=2^{7} \cdot 3^{4} \cdot 5 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |