Group action invariants
| Degree $n$: | $18$ | |
| Transitive number $t$: | $16$ | |
| Group: | $C_9\times S_3$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $9$ | |
| Generators: | (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,12)(8,10)(9,11), (1,18,9,13,4,11,2,16,7,14,5,12,3,17,8,15,6,10) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $9$: $C_9$ $18$: $S_3\times C_3$, $C_{18}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 9: None
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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $(10,11,12)(13,14,15)(16,17,18)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $(10,12,11)(13,15,14)(16,18,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$ |
| $ 9, 9 $ | $1$ | $9$ | $( 1, 4, 7, 3, 6, 9, 2, 5, 8)(10,13,16,12,15,18,11,14,17)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 4, 7, 3, 6, 9, 2, 5, 8)(10,14,18,12,13,17,11,15,16)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 4, 7, 3, 6, 9, 2, 5, 8)(10,15,17,12,14,16,11,13,18)$ |
| $ 9, 9 $ | $1$ | $9$ | $( 1, 5, 9, 3, 4, 8, 2, 6, 7)(10,14,18,12,13,17,11,15,16)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 5, 9, 3, 4, 8, 2, 6, 7)(10,15,17,12,14,16,11,13,18)$ |
| $ 9, 9 $ | $1$ | $9$ | $( 1, 6, 8, 3, 5, 7, 2, 4, 9)(10,15,17,12,14,16,11,13,18)$ |
| $ 9, 9 $ | $1$ | $9$ | $( 1, 7, 6, 2, 8, 4, 3, 9, 5)(10,16,15,11,17,13,12,18,14)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 7, 6, 2, 8, 4, 3, 9, 5)(10,17,14,11,18,15,12,16,13)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 7, 6, 2, 8, 4, 3, 9, 5)(10,18,13,11,16,14,12,17,15)$ |
| $ 9, 9 $ | $1$ | $9$ | $( 1, 8, 5, 2, 9, 6, 3, 7, 4)(10,17,14,11,18,15,12,16,13)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 8, 5, 2, 9, 6, 3, 7, 4)(10,18,13,11,16,14,12,17,15)$ |
| $ 9, 9 $ | $1$ | $9$ | $( 1, 9, 4, 2, 7, 5, 3, 8, 6)(10,18,13,11,16,14,12,17,15)$ |
| $ 18 $ | $3$ | $18$ | $( 1,10, 4,13, 7,16, 3,12, 6,15, 9,18, 2,11, 5,14, 8,17)$ |
| $ 18 $ | $3$ | $18$ | $( 1,10, 5,14, 9,18, 3,12, 4,13, 8,17, 2,11, 6,15, 7,16)$ |
| $ 18 $ | $3$ | $18$ | $( 1,10, 6,15, 8,17, 3,12, 5,14, 7,16, 2,11, 4,13, 9,18)$ |
| $ 6, 6, 6 $ | $3$ | $6$ | $( 1,13, 3,15, 2,14)( 4,16, 6,18, 5,17)( 7,12, 9,11, 8,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,12)( 8,10)( 9,11)$ |
| $ 6, 6, 6 $ | $3$ | $6$ | $( 1,13, 2,14, 3,15)( 4,16, 5,17, 6,18)( 7,12, 8,10, 9,11)$ |
| $ 18 $ | $3$ | $18$ | $( 1,16, 7,15, 6,11, 2,17, 8,13, 4,12, 3,18, 9,14, 5,10)$ |
| $ 18 $ | $3$ | $18$ | $( 1,16, 8,13, 5,10, 2,17, 9,14, 6,11, 3,18, 7,15, 4,12)$ |
| $ 18 $ | $3$ | $18$ | $( 1,16, 9,14, 4,12, 2,17, 7,15, 5,10, 3,18, 8,13, 6,11)$ |
Group invariants
| Order: | $54=2 \cdot 3^{3}$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [54, 4] |
| Character table: not available. |