Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $199$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,14,8,18,11,6,3,15,10)(2,13,7,17,12,5,4,16,9), (1,16,8,4,11,9,18,13,6,2,15,7,3,12,10,17,14,5), (1,13,7,17,15,6,3,12,9,2,14,8,18,16,5,4,11,10) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 12: $A_4$ 24: $A_4\times C_2$ 324: $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4\times C_2$
Low degree siblings
18T199, 18T201, 18T205Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
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, 3, 3, 3 $ | $4$ | $3$ | $( 1,18, 3)( 2,17, 4)( 5, 9, 7)( 6,10, 8)(11,15,14)(12,16,13)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 3,18)( 2, 4,17)( 5, 7, 9)( 6, 8,10)(11,14,15)(12,13,16)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 5, 9, 7)( 6,10, 8)(11,15,14)(12,16,13)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 1, 3,18)( 2, 4,17)$ |
| $ 6, 2, 2, 2, 2, 2, 1, 1 $ | $54$ | $6$ | $( 1, 4)( 2, 3)( 5, 9)( 6,10)(11,13,15,12,14,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 3,17)( 4,18)( 5, 7)( 6, 8)(11,12)(13,14)(15,16)$ |
| $ 9, 9 $ | $36$ | $9$ | $( 1,14, 8,18,11, 6, 3,15,10)( 2,13, 7,17,12, 5, 4,16, 9)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $36$ | $3$ | $( 1,11,10)( 2,12, 9)( 3,14, 6)( 4,13, 5)( 7,17,16)( 8,18,15)$ |
| $ 9, 9 $ | $36$ | $9$ | $( 1,15, 6, 3,11, 8,18,14,10)( 2,16, 5, 4,12, 7,17,13, 9)$ |
| $ 9, 9 $ | $36$ | $9$ | $( 1, 8,11, 3,10,14,18, 6,15)( 2, 7,12, 4, 9,13,17, 5,16)$ |
| $ 9, 9 $ | $36$ | $9$ | $( 1, 6,14,18,10,11, 3, 8,15)( 2, 5,13,17, 9,12, 4, 7,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $36$ | $3$ | $( 1,10,15)( 2, 9,16)( 3, 6,11)( 4, 5,12)( 7,13,17)( 8,14,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 7, 9)( 8,10)(11,13)(12,14)(15,16)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 1, 1 $ | $54$ | $6$ | $( 1,18, 3)( 2,17, 4)( 5, 9)( 6,10)(11,12)(13,15)(14,16)$ |
| $ 6, 6, 6 $ | $4$ | $6$ | $( 1, 4,18, 2, 3,17)( 5,10, 7, 6, 9, 8)(11,16,14,12,15,13)$ |
| $ 6, 6, 2, 2, 2 $ | $12$ | $6$ | $( 1, 2)( 3, 4)( 5, 8, 9, 6, 7,10)(11,13,15,12,14,16)(17,18)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $6$ | $6$ | $( 1,17, 3, 2,18, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 6, 6, 6 $ | $4$ | $6$ | $( 1,17, 3, 2,18, 4)( 5,10, 7, 6, 9, 8)(11,16,14,12,15,13)$ |
| $ 18 $ | $36$ | $18$ | $( 1,14, 5, 4,16,10,18,11, 7, 2,13, 6, 3,15, 9,17,12, 8)$ |
| $ 6, 6, 6 $ | $36$ | $6$ | $( 1,11, 5, 2,12, 6)( 3,14, 9, 4,13,10)( 7,17,16, 8,18,15)$ |
| $ 18 $ | $36$ | $18$ | $( 1,15, 5,17,13, 8, 3,11, 9, 2,16, 6,18,14, 7, 4,12,10)$ |
| $ 18 $ | $36$ | $18$ | $( 1, 8,14, 4, 9,12,18, 6,15, 2, 7,13, 3,10,11,17, 5,16)$ |
| $ 6, 6, 6 $ | $36$ | $6$ | $( 1, 6,14, 2, 5,13)( 3, 8,11, 4, 7,12)( 9,16,18,10,15,17)$ |
| $ 18 $ | $36$ | $18$ | $( 1,10,14,17, 7,16, 3, 6,11, 2, 9,13,18, 8,15, 4, 5,12)$ |
Group invariants
| Order: | $648=2^{3} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [648, 706] |
| Character table: Data not available. |