Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $486$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,5)(2,6)(7,17)(8,18)(9,13,11,15)(10,14,12,16), (1,12,16,2,11,15)(3,10,14,6,7,17)(4,9,13,5,8,18) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 6: $S_3$ 12: $D_{6}$ 24: $S_4$ x 3 48: $S_4\times C_2$ x 3 96: $V_4^2:S_3$ 192: 12T100 1296: $S_3\wr S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: $S_4\times C_2$
Degree 9: $S_3\wr S_3$
Low degree siblings
18T483 x 2, 18T484 x 2, 18T486, 18T488 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, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 7,11, 9)( 8,12,10)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 7,11, 9)( 8,12,10)(13,17,15)(14,18,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,11, 9)( 8,12,10)(13,17,15)(14,18,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 9,11)(10,12)(15,17)(16,18)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $54$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 9,11)(10,12)(15,17)(16,18)$ |
| $ 6, 2, 2, 2, 2, 2, 1, 1 $ | $108$ | $6$ | $( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,12)(13,18,15,14,17,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $54$ | $2$ | $( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 7,10)( 8, 9)(11,12)(13,16)(14,15)(17,18)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $54$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7,10)( 8, 9)(11,12)(13,16)(14,15)(17,18)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $12$ | $6$ | $( 7,12, 9, 8,11,10)(13,18,15,14,17,16)$ |
| $ 6, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $12$ | $6$ | $( 7, 8)( 9,10)(11,12)(13,18,15,14,17,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 6, 6, 3, 3 $ | $24$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7,12, 9, 8,11,10)(13,18,15,14,17,16)$ |
| $ 6, 3, 3, 2, 2, 2 $ | $24$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7, 8)( 9,10)(11,12)(13,18,15,14,17,16)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $6$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $288$ | $3$ | $( 1,17, 7)( 2,18, 8)( 3,15,11)( 4,16,12)( 5,13, 9)( 6,14,10)$ |
| $ 9, 9 $ | $576$ | $9$ | $( 1,17,11, 3,15, 9, 5,13, 7)( 2,18,12, 4,16,10, 6,14, 8)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $72$ | $6$ | $( 1,13, 3,15, 5,17)( 2,14, 4,16, 6,18)$ |
| $ 6, 6, 3, 3 $ | $144$ | $6$ | $( 1,13, 3,15, 5,17)( 2,14, 4,16, 6,18)( 7,11, 9)( 8,12,10)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $2$ | $( 1,17)( 2,18)( 3,13)( 4,14)( 5,15)( 6,16)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $72$ | $6$ | $( 1,17)( 2,18)( 3,13)( 4,14)( 5,15)( 6,16)( 7,11, 9)( 8,12,10)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $324$ | $4$ | $( 1,13, 3,17)( 2,14, 4,18)( 5,15)( 6,16)( 9,11)(10,12)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $324$ | $4$ | $( 1,18, 2,17)( 3,14, 6,15)( 4,13, 5,16)( 7,10)( 8, 9)(11,12)$ |
| $ 12, 6 $ | $144$ | $12$ | $( 1,16, 4,13, 5,18, 2,15, 3,14, 6,17)( 7,12, 9, 8,11,10)$ |
| $ 12, 2, 2, 2 $ | $72$ | $12$ | $( 1,16, 4,13, 5,18, 2,15, 3,14, 6,17)( 7, 8)( 9,10)(11,12)$ |
| $ 6, 4, 4, 4 $ | $72$ | $12$ | $( 1,18, 2,17)( 3,16, 4,15)( 5,14, 6,13)( 7,12, 9, 8,11,10)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $36$ | $4$ | $( 1,18, 2,17)( 3,16, 4,15)( 5,14, 6,13)( 7, 8)( 9,10)(11,12)$ |
| $ 6, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $6$ | $( 1, 4, 5, 2, 3, 6)(15,17)(16,18)$ |
| $ 6, 3, 3, 2, 2, 1, 1 $ | $72$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7,11, 9)( 8,12,10)(15,17)(16,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)(15,17)(16,18)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,11, 9)( 8,12,10)(15,17)(16,18)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)$ |
| $ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $6$ | $( 1, 6)( 2, 5)( 3, 4)( 7,11, 9)( 8,12,10)$ |
| $ 3, 3, 3, 3, 2, 2, 2 $ | $36$ | $6$ | $( 1, 6)( 2, 5)( 3, 4)( 7,11, 9)( 8,12,10)(13,17,15)(14,18,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $81$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)( 9,11)(10,12)(15,17)(16,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $27$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9)(11,12)(13,18)(14,17)(15,16)$ |
| $ 6, 6, 2, 2, 2 $ | $36$ | $6$ | $( 1, 6)( 2, 5)( 3, 4)( 7,12, 9, 8,11,10)(13,16,17,14,15,18)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $36$ | $6$ | $( 1, 6)( 2, 5)( 3, 4)( 7, 8)( 9,10)(11,12)(13,16,17,14,15,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 6, 6, 6 $ | $864$ | $6$ | $( 1,16, 8, 2,15, 7)( 3,14,10, 6,17,11)( 4,13, 9, 5,18,12)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $108$ | $4$ | $( 1,16, 2,15)( 3,18, 6,13)( 4,17, 5,14)$ |
| $ 4, 4, 4, 3, 3 $ | $216$ | $12$ | $( 1,16, 2,15)( 3,18, 6,13)( 4,17, 5,14)( 7,11, 9)( 8,12,10)$ |
| $ 12, 2, 2, 1, 1 $ | $216$ | $12$ | $( 1,18, 6,13, 3,16, 2,17, 5,14, 4,15)( 9,11)(10,12)$ |
| $ 4, 4, 4, 2, 2, 1, 1 $ | $108$ | $4$ | $( 1,16, 2,15)( 3,14, 4,13)( 5,18, 6,17)( 9,11)(10,12)$ |
| $ 6, 6, 2, 2, 2 $ | $216$ | $6$ | $( 1,13, 5,17, 3,15)( 2,14, 6,18, 4,16)( 7,10)( 8, 9)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $108$ | $2$ | $( 1,15)( 2,16)( 3,17)( 4,18)( 5,13)( 6,14)( 7,10)( 8, 9)(11,12)$ |
| $ 6, 4, 4, 2, 2 $ | $216$ | $12$ | $( 1,13, 5,15)( 2,14, 6,16)( 3,17)( 4,18)( 7,12, 9, 8,11,10)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $108$ | $4$ | $( 1,13, 5,15)( 2,14, 6,16)( 3,17)( 4,18)( 7, 8)( 9,10)(11,12)$ |
Group invariants
| Order: | $5184=2^{6} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |