Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $518$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,17,11,5,13,9)(2,18,12,6,14,10)(3,16,8,4,15,7), (3,4)(5,6)(7,18)(8,17)(9,13,10,14)(11,15,12,16) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 6: $S_3$ x 2 12: $D_{6}$ x 2 36: $S_3^2$ 108: $C_3^2 : D_{6} $ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: None
Degree 9: $C_3^2 : D_{6} $
Low degree siblings
12T268 x 2, 18T517 x 2, 18T518, 18T519 x 2, 18T520 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 7, 8)(11,12)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(13,14)(17,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $96$ | $3$ | $( 1,11,13)( 2,12,14)( 3, 8,15)( 4, 7,16)( 5, 9,17)( 6,10,18)$ |
| $ 6, 6, 3, 3 $ | $288$ | $6$ | $( 1,12,14, 2,11,13)( 3, 7,15)( 4, 8,16)( 5, 9,18, 6,10,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $128$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9,12)( 8,10,11)(13,15,18)(14,16,17)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $96$ | $3$ | $( 7,11, 9)( 8,12,10)(13,15,18)(14,16,17)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $288$ | $6$ | $( 1, 2)( 5, 6)( 7,11,10)( 8,12, 9)(13,16,17)(14,15,18)$ |
| $ 6, 6, 3, 3 $ | $576$ | $6$ | $( 1,11,17, 2,12,18)( 3, 8,14, 4, 7,13)( 5, 9,16)( 6,10,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $192$ | $3$ | $( 1,12,17)( 2,11,18)( 3, 8,13)( 4, 7,14)( 5,10,15)( 6, 9,16)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $72$ | $4$ | $( 3, 6, 4, 5)( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 2, 2, 2, 2, 2, 2, 1, 1 $ | $216$ | $4$ | $( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,11)(10,12)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ | $216$ | $4$ | $( 3, 5, 4, 6)( 9,11,10,12)(13,16)(14,15)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $72$ | $2$ | $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,11)(10,12)(13,15)(14,16)(17,18)$ |
| $ 6, 6, 6 $ | $576$ | $6$ | $( 1,11,18, 6, 7,16)( 2,12,17, 5, 8,15)( 3,10,14, 4, 9,13)$ |
| $ 12, 3, 3 $ | $576$ | $12$ | $( 1,11,17, 6, 7,15, 2,12,18, 5, 8,16)( 3,10,14)( 4, 9,13)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $108$ | $4$ | $( 3, 4)( 5, 6)( 7,18)( 8,17)( 9,13,10,14)(11,15,12,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $108$ | $4$ | $( 7,17, 8,18)( 9,13,10,14)(11,16)(12,15)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $216$ | $4$ | $( 1, 2)( 5, 6)( 7,17)( 8,18)( 9,14,10,13)(11,15,12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $108$ | $2$ | $( 1, 2)( 3, 4)( 7,17)( 8,18)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $2$ | $( 7,18)( 8,17)( 9,14)(10,13)(11,16)(12,15)$ |
| $ 6, 6, 3, 3 $ | $1152$ | $6$ | $( 1,11, 3, 7, 6, 9)( 2,12, 4, 8, 5,10)(13,18,16)(14,17,15)$ |
| $ 4, 2, 2, 2, 2, 2, 2, 1, 1 $ | $72$ | $4$ | $( 3, 5, 4, 6)( 7,18)( 8,17)( 9,15)(10,16)(11,14)(12,13)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $216$ | $4$ | $( 1, 2)( 3, 6)( 4, 5)( 7,18, 8,17)( 9,16)(10,15)(11,13,12,14)$ |
| $ 4, 4, 4, 2, 2, 1, 1 $ | $216$ | $4$ | $( 3, 5, 4, 6)( 7,17)( 8,18)( 9,16,10,15)(11,14,12,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $72$ | $2$ | $( 1, 2)( 3, 5)( 4, 6)( 7,18)( 8,17)( 9,16)(10,15)(11,13)(12,14)$ |
| $ 6, 6, 4, 1, 1 $ | $576$ | $12$ | $( 3, 5, 4, 6)( 7,13,10,17,12,15)( 8,14, 9,18,11,16)$ |
| $ 6, 6, 2, 2, 2 $ | $576$ | $6$ | $( 1, 2)( 3, 6)( 4, 5)( 7,14,10,17,12,16)( 8,13, 9,18,11,15)$ |
Group invariants
| Order: | $6912=2^{8} \cdot 3^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |