Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $394$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,10,11,3,6,16)(2,9,12,4,5,15)(7,14,17,8,13,18), (1,8,15,17,10,11)(2,7,16,18,9,12)(3,6,14)(4,5,13), (1,6)(2,5)(3,10)(4,9)(7,17)(8,18)(11,14)(12,13)(15,16) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_2^2$ x 7 6: $S_3$ 8: $C_2^3$ 12: $D_{6}$ x 3 24: $S_4$, $S_3 \times C_2^2$ 48: $S_4\times C_2$ x 3 96: 12T48 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
18T394 x 3, 18T397 x 4Siblings 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 $ | $6$ | $3$ | $( 5,10, 8)( 6, 9, 7)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 5,10, 8)( 6, 9, 7)(11,13,15)(12,14,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5,10, 8)( 6, 9, 7)(11,13,15)(12,14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1,18)( 2,17)( 3, 4)( 5, 7)( 6, 8)( 9,10)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $54$ | $6$ | $( 1,18)( 2,17)( 3, 4)( 5, 7)( 6, 8)( 9,10)(11,13,15)(12,14,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $72$ | $3$ | $( 1,11, 6)( 2,12, 5)( 3,16,10)( 4,15, 9)( 7,17,13)( 8,18,14)$ |
| $ 9, 9 $ | $144$ | $9$ | $( 1,11, 9, 4,15, 7,17,13, 6)( 2,12,10, 3,16, 8,18,14, 5)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 3)( 2, 4)(17,18)$ |
| $ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $6$ | $( 1, 3)( 2, 4)( 5,10, 8)( 6, 9, 7)(17,18)$ |
| $ 3, 3, 3, 3, 2, 2, 2 $ | $36$ | $6$ | $( 1, 3)( 2, 4)( 5,10, 8)( 6, 9, 7)(11,13,15)(12,14,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $27$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,16)(14,15)(17,18)$ |
| $ 6, 6, 6 $ | $216$ | $6$ | $( 1,16,10, 3,11, 6)( 2,15, 9, 4,12, 5)( 7,17,14, 8,18,13)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3,18)( 4,17)( 7, 9)( 8,10)(13,15)(14,16)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1,18, 4, 2,17, 3)( 5, 7,10, 6, 8, 9)(13,15)(14,16)$ |
| $ 6, 2, 2, 2, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1,18, 4, 2,17, 3)( 5, 6)( 7, 8)( 9,10)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,15)(14,16)(17,18)$ |
| $ 6, 6, 3, 3 $ | $216$ | $6$ | $( 1,11, 6)( 2,12, 5)( 3,14,10,18,16, 8)( 4,13, 9,17,15, 7)$ |
| $ 6, 2, 2, 2, 2, 1, 1, 1, 1 $ | $54$ | $6$ | $( 1, 3,17, 2, 4,18)( 7, 9)( 8,10)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 3, 4)( 7, 9)( 8,10)(13,15)(14,16)(17,18)$ |
| $ 6, 6, 2, 2, 2 $ | $12$ | $6$ | $( 1, 3,17, 2, 4,18)( 5, 7,10, 6, 8, 9)(11,12)(13,14)(15,16)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $6$ | $6$ | $( 1, 3,17, 2, 4,18)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 6, 6, 6 $ | $8$ | $6$ | $( 1, 3,17, 2, 4,18)( 5, 7,10, 6, 8, 9)(11,14,15,12,13,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 18 $ | $144$ | $18$ | $( 1,16, 8, 3,13, 9,17,12, 5, 2,15, 7, 4,14,10,18,11, 6)$ |
| $ 6, 6, 6 $ | $72$ | $6$ | $( 1,16, 5, 2,15, 6)( 3,13, 7, 4,14, 8)( 9,17,12,10,18,11)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $54$ | $4$ | $( 1,11,18,14)( 2,12,17,13)( 3,16, 4,15)$ |
| $ 4, 4, 4, 3, 3 $ | $108$ | $12$ | $( 1,11,18,14)( 2,12,17,13)( 3,16, 4,15)( 5,10, 8)( 6, 9, 7)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $54$ | $2$ | $( 1,11)( 2,12)( 3,16)( 4,15)( 5, 7)( 6, 8)( 9,10)(13,17)(14,18)$ |
| $ 6, 6, 2, 2, 2 $ | $108$ | $6$ | $( 1,13,17,15, 4,11)( 2,14,18,16, 3,12)( 5, 7)( 6, 8)( 9,10)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $36$ | $6$ | $( 1,11,17,13, 4,15)( 2,12,18,14, 3,16)$ |
| $ 6, 6, 3, 3 $ | $72$ | $6$ | $( 1,11,17,13, 4,15)( 2,12,18,14, 3,16)( 5,10, 8)( 6, 9, 7)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 1,15)( 2,16)( 3,14)( 4,13)(11,17)(12,18)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $36$ | $6$ | $( 1,15)( 2,16)( 3,14)( 4,13)( 5,10, 8)( 6, 9, 7)(11,17)(12,18)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $162$ | $4$ | $( 1,11, 2,12)( 3,16,17,13)( 4,15,18,14)( 5, 7)( 6, 8)( 9,10)$ |
| $ 4, 4, 4, 2, 2, 1, 1 $ | $162$ | $4$ | $( 1,11, 3,14)( 2,12, 4,13)( 7, 9)( 8,10)(15,18,16,17)$ |
| $ 6, 6, 6 $ | $72$ | $6$ | $( 1,11, 4,13,17,15)( 2,12, 3,14,18,16)( 5, 7,10, 6, 8, 9)$ |
| $ 6, 6, 2, 2, 2 $ | $36$ | $6$ | $( 1,11, 4,13,17,15)( 2,12, 3,14,18,16)( 5, 6)( 7, 8)( 9,10)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $36$ | $6$ | $( 1,15)( 2,16)( 3,12)( 4,11)( 5, 7,10, 6, 8, 9)(13,17)(14,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $18$ | $2$ | $( 1,15)( 2,16)( 3,12)( 4,11)( 5, 6)( 7, 8)( 9,10)(13,17)(14,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $54$ | $2$ | $( 1,11)( 2,12)( 3,14)( 4,13)( 7, 9)( 8,10)(15,17)(16,18)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $108$ | $6$ | $( 1,13, 4,15,17,11)( 2,14, 3,16,18,12)( 7, 9)( 8,10)$ |
| $ 6, 4, 4, 4 $ | $108$ | $12$ | $( 1,11,18,16)( 2,12,17,15)( 3,14, 4,13)( 5, 7,10, 6, 8, 9)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $54$ | $4$ | $( 1,11,18,16)( 2,12,17,15)( 3,14, 4,13)( 5, 6)( 7, 8)( 9,10)$ |
Group invariants
| Order: | $2592=2^{5} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |