Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $483$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (7,8)(9,12)(10,11)(13,16,17,14,15,18), (1,9,17,5,11,13,3,7,15)(2,10,18,6,12,14,4,8,16), (1,6,3,2,5,4)(7,14,12,15,9,18,8,13,11,16,10,17) | |
| $|\Aut(F/K)|$: | $2$ |
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: $S_4$
Degree 9: $S_3\wr S_3$
Low degree siblings
18T483, 18T484 x 2, 18T486 x 2, 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 $ | $6$ | $3$ | $(13,17,15)(14,18,16)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)$ |
| $ 6, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $6$ | $( 7, 8)( 9,12)(10,11)(13,16,17,14,15,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,18)$ |
| $ 6, 3, 3, 2, 2, 2 $ | $72$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7, 8)( 9,12)(10,11)(13,16,17,14,15,18)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $36$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,18)$ |
| $ 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 $ | $54$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 9,11)(10,12)(15,17)(16,18)$ |
| $ 6, 2, 2, 2, 2, 2, 1, 1 $ | $108$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7, 8)( 9,12)(10,11)(15,17)(16,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $54$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(15,17)(16,18)$ |
| $ 6, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $12$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7, 8)( 9,10)(11,12)$ |
| $ 6, 3, 3, 2, 2, 2 $ | $24$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7, 8)( 9,10)(11,12)(13,17,15)(14,18,16)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $12$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7,10,11, 8, 9,12)$ |
| $ 6, 6, 3, 3 $ | $24$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7,10,11, 8, 9,12)(13,17,15)(14,18,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $6$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,17,15)(14,18,16)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 9,11)(10,12)(13,16,17,14,15,18)$ |
| $ 6, 2, 2, 2, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 9,11)(10,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 9,11)(10,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 5)( 4, 6)$ |
| $ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $6$ | $( 3, 5)( 4, 6)(13,17,15)(14,18,16)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $36$ | $6$ | $( 3, 5)( 4, 6)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3, 5)( 4, 6)( 9,11)(10,12)(15,17)(16,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $81$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)( 7, 8)( 9,12)(10,11)(15,17)(16,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)(13,16)(14,15)(17,18)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $54$ | $6$ | $( 1, 6)( 2, 5)( 3, 4)( 7, 9,11)( 8,10,12)(13,16)(14,15)(17,18)$ |
| $ 9, 9 $ | $576$ | $9$ | $( 1, 9,17, 5,11,13, 3, 7,15)( 2,10,18, 6,12,14, 4, 8,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $288$ | $3$ | $( 1, 9,15)( 2,10,16)( 3, 7,13)( 4, 8,14)( 5,11,17)( 6,12,18)$ |
| $ 6, 6, 3, 3 $ | $864$ | $6$ | $( 1,12,15)( 2,11,16)( 3, 8,17, 5,10,13)( 4, 7,18, 6, 9,14)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $324$ | $4$ | $( 1,12)( 2,11)( 3,10, 5, 8)( 4, 9, 6, 7)(15,17)(16,18)$ |
| $ 12, 2, 2, 2 $ | $216$ | $12$ | $( 1, 9, 6, 8, 3,11, 2,10, 5, 7, 4,12)(13,16)(14,15)(17,18)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $108$ | $4$ | $( 1,11, 2,12)( 3, 7, 4, 8)( 5, 9, 6,10)(13,16)(14,15)(17,18)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $72$ | $6$ | $( 1,10, 5, 8, 3,12)( 2, 9, 6, 7, 4,11)$ |
| $ 6, 6, 3, 3 $ | $144$ | $6$ | $( 1,10, 5, 8, 3,12)( 2, 9, 6, 7, 4,11)(13,17,15)(14,18,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $2$ | $( 1,12)( 2,11)( 3, 8)( 4, 7)( 5,10)( 6, 9)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $72$ | $6$ | $( 1,12)( 2,11)( 3, 8)( 4, 7)( 5,10)( 6, 9)(13,17,15)(14,18,16)$ |
| $ 6, 4, 4, 4 $ | $216$ | $12$ | $( 1,11, 2,12)( 3, 9, 6, 8)( 4,10, 5, 7)(13,16,17,14,15,18)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $108$ | $4$ | $( 1,11, 2,12)( 3, 9, 6, 8)( 4,10, 5, 7)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $108$ | $2$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(15,17)(16,18)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $216$ | $6$ | $( 1, 8, 5,10, 3,12)( 2, 7, 6, 9, 4,11)(15,17)(16,18)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $324$ | $4$ | $( 1, 9, 4,12)( 2,10, 3,11)( 5, 7, 6, 8)(13,16)(14,15)(17,18)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $108$ | $4$ | $( 1,10, 3,12)( 2, 9, 4,11)( 5, 8)( 6, 7)$ |
| $ 4, 4, 3, 3, 2, 2 $ | $216$ | $12$ | $( 1,10, 3,12)( 2, 9, 4,11)( 5, 8)( 6, 7)(13,17,15)(14,18,16)$ |
| $ 6, 4, 4, 4 $ | $72$ | $12$ | $( 1,11, 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,16,17,14,15,18)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $36$ | $4$ | $( 1,11, 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,14)(15,16)(17,18)$ |
| $ 12, 6 $ | $144$ | $12$ | $( 1, 7, 6,10, 3,11, 2, 8, 5, 9, 4,12)(13,16,17,14,15,18)$ |
| $ 12, 2, 2, 2 $ | $72$ | $12$ | $( 1, 7, 6,10, 3,11, 2, 8, 5, 9, 4,12)(13,14)(15,16)(17,18)$ |
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. |