Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $285$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,4,18,2,3,17)(7,9)(8,10)(11,13,15,12,14,16), (1,16,5,17,11,8)(2,15,6,18,12,7)(3,13,9,4,14,10) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $C_6$ x 3 12: $A_4$, $C_6\times C_2$ 24: $A_4\times C_2$ x 3 48: $C_2^2 \times A_4$ 648: $S_3 \wr C_3 $ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4\times C_2$
Degree 9: $S_3 \wr C_3 $
Low degree siblings
18T283 x 4, 18T285, 18T286 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, 3, 3, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 1,18, 3)( 2,17, 4)(11,15,14)(12,16,13)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3,18)( 2, 4,17)( 5, 9, 7)( 6,10, 8)(11,15,14)(12,16,13)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $(11,15,14)(12,16,13)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1, 4,18, 2, 3,17)( 7, 9)( 8,10)(11,13,15,12,14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 7, 9)( 8,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 6, 2, 2, 2, 2, 2, 1, 1 $ | $18$ | $6$ | $( 1, 2)( 3, 4)( 5, 9)( 6,10)(11,13,15,12,14,16)(17,18)$ |
| $ 6, 2, 2, 2, 2, 2, 1, 1 $ | $18$ | $6$ | $( 1,17, 3, 2,18, 4)( 5, 9)( 6,10)(11,12)(13,14)(15,16)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $54$ | $6$ | $( 1, 4)( 2, 3)( 5,10)( 6, 9)( 7, 8)(11,15,14)(12,16,13)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1,17)( 2,18)( 3, 4)( 5,10)( 6, 9)( 7, 8)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3,18)( 4,17)( 7, 9)( 8,10)(13,16)(14,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $36$ | $3$ | $( 1, 5,11)( 2, 6,12)( 3, 9,14)( 4,10,13)( 7,15,18)( 8,16,17)$ |
| $ 9, 9 $ | $72$ | $9$ | $( 1, 5,15, 3, 9,11,18, 7,14)( 2, 6,16, 4,10,12,17, 8,13)$ |
| $ 6, 6, 3, 3 $ | $108$ | $6$ | $( 1, 5,13,18, 9,16)( 2, 6,14,17,10,15)( 3, 7,12)( 4, 8,11)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $36$ | $3$ | $( 1,11, 5)( 2,12, 6)( 3,14, 9)( 4,13,10)( 7,18,15)( 8,17,16)$ |
| $ 9, 9 $ | $72$ | $9$ | $( 1,15, 7, 3,11, 5,18,14, 9)( 2,16, 8, 4,12, 6,17,13,10)$ |
| $ 6, 6, 3, 3 $ | $108$ | $6$ | $( 1,13, 8)( 2,14, 7)( 3,16,10,18,12, 6)( 4,15, 9,17,11, 5)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(11,13)(12,14)(15,16)$ |
| $ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 1,18, 3)( 2,17, 4)(11,12)(13,15)(14,16)$ |
| $ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 5, 9, 7)( 6,10, 8)(11,16)(12,15)(13,14)$ |
| $ 3, 3, 3, 3, 2, 2, 2 $ | $36$ | $6$ | $( 1,18, 3)( 2,17, 4)( 5, 9, 7)( 6,10, 8)(11,13)(12,14)(15,16)$ |
| $ 6, 2, 2, 2, 2, 1, 1, 1, 1 $ | $54$ | $6$ | $( 1, 4,18, 2, 3,17)( 7, 9)( 8,10)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 3, 4)( 7, 9)( 8,10)(11,14)(12,13)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $27$ | $2$ | $( 1, 4)( 2, 3)( 5,10)( 6, 9)( 7, 8)(11,12)(13,15)(14,16)(17,18)$ |
| $ 6, 6, 6 $ | $8$ | $6$ | $( 1, 4,18, 2, 3,17)( 5,10, 7, 6, 9, 8)(11,13,15,12,14,16)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $6$ | $6$ | $( 1, 2)( 3, 4)( 5,10, 7, 6, 9, 8)(11,12)(13,14)(15,16)(17,18)$ |
| $ 6, 6, 2, 2, 2 $ | $12$ | $6$ | $( 1, 2)( 3, 4)( 5, 8, 9, 6, 7,10)(11,13,15,12,14,16)(17,18)$ |
| $ 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 $ | $108$ | $6$ | $( 1, 5,11, 4,10,13)( 2, 6,12, 3, 9,14)( 7,15,17, 8,16,18)$ |
| $ 18 $ | $72$ | $18$ | $( 1, 5,13, 4, 8,11,18, 9,16, 2, 6,14, 3, 7,12,17,10,15)$ |
| $ 6, 6, 6 $ | $36$ | $6$ | $( 1, 7,16, 2, 8,15)( 3, 9,13, 4,10,14)( 5,12,17, 6,11,18)$ |
| $ 6, 6, 6 $ | $108$ | $6$ | $( 1,11,10, 4,13, 5)( 2,12, 9, 3,14, 6)( 7,18,15, 8,17,16)$ |
| $ 18 $ | $72$ | $18$ | $( 1,13, 5, 4,15,10,18,12, 7, 2,14, 6, 3,16, 9,17,11, 8)$ |
| $ 6, 6, 6 $ | $36$ | $6$ | $( 1,16, 9, 2,15,10)( 3,12, 7, 4,11, 8)( 5,17,14, 6,18,13)$ |
Group invariants
| Order: | $1296=2^{4} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [1296, 3494] |
| Character table: Data not available. |