Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $119$ | |
| Group : | $C_2\times C_3\wr S_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,3,17,2,4,18)(5,9,7,6,10,8)(11,16,13,12,15,14), (1,8,16,17,6,14,4,9,12)(2,7,15,18,5,13,3,10,11), (1,16,17,14,4,12)(2,15,18,13,3,11) | |
| $|\Aut(F/K)|$: | $6$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $S_3$, $C_6$ x 3 12: $D_{6}$, $C_6\times C_2$ 18: $S_3\times C_3$ 36: $C_6\times S_3$ 54: $C_3^2 : C_6$ 108: 18T41 162: $C_3 \wr S_3 $ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $D_{6}$
Degree 9: $C_3 \wr S_3 $
Low degree siblings
18T119 x 5Siblings 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 $ | $3$ | $3$ | $(11,13,15)(12,14,16)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $(11,15,13)(12,16,14)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 5, 7,10)( 6, 8, 9)(11,13,15)(12,14,16)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 5, 7,10)( 6, 8, 9)(11,15,13)(12,16,14)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 5,10, 7)( 6, 9, 8)(11,15,13)(12,16,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5,11)( 6,12)( 7,13)( 8,14)( 9,16)(10,15)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $9$ | $6$ | $( 5,11, 7,13,10,15)( 6,12, 8,14, 9,16)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $9$ | $6$ | $( 5,11,10,15, 7,13)( 6,12, 9,16, 8,14)$ |
| $ 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, 2, 2, 2, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,14,15,12,13,16)(17,18)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,16,13,12,15,14)(17,18)$ |
| $ 6, 6, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 8,10, 6, 7, 9)(11,14,15,12,13,16)(17,18)$ |
| $ 6, 6, 2, 2, 2 $ | $6$ | $6$ | $( 1, 2)( 3, 4)( 5, 8,10, 6, 7, 9)(11,16,13,12,15,14)(17,18)$ |
| $ 6, 6, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 9, 7, 6,10, 8)(11,16,13,12,15,14)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5,12)( 6,11)( 7,14)( 8,13)( 9,15)(10,16)(17,18)$ |
| $ 6, 6, 2, 2, 2 $ | $9$ | $6$ | $( 1, 2)( 3, 4)( 5,12, 7,14,10,16)( 6,11, 8,13, 9,15)(17,18)$ |
| $ 6, 6, 2, 2, 2 $ | $9$ | $6$ | $( 1, 2)( 3, 4)( 5,12,10,16, 7,14)( 6,11, 9,15, 8,13)(17,18)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1, 3,17, 2, 4,18)( 5, 8,10, 6, 7, 9)(11,14,15,12,13,16)$ |
| $ 6, 6, 6 $ | $3$ | $6$ | $( 1, 3,17, 2, 4,18)( 5, 8,10, 6, 7, 9)(11,16,13,12,15,14)$ |
| $ 6, 6, 6 $ | $3$ | $6$ | $( 1, 3,17, 2, 4,18)( 5, 9, 7, 6,10, 8)(11,16,13,12,15,14)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $9$ | $6$ | $( 1, 3,17, 2, 4,18)( 5,12)( 6,11)( 7,14)( 8,13)( 9,15)(10,16)$ |
| $ 6, 6, 6 $ | $9$ | $6$ | $( 1, 3,17, 2, 4,18)( 5,12, 7,14,10,16)( 6,11, 8,13, 9,15)$ |
| $ 6, 6, 6 $ | $9$ | $6$ | $( 1, 3,17, 2, 4,18)( 5,12,10,16, 7,14)( 6,11, 9,15, 8,13)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 7,10)( 6, 8, 9)(11,13,15)(12,14,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 7,10)( 6, 8, 9)(11,15,13)(12,16,14)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5,10, 7)( 6, 9, 8)(11,15,13)(12,16,14)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $9$ | $6$ | $( 1, 4,17)( 2, 3,18)( 5,11)( 6,12)( 7,13)( 8,14)( 9,16)(10,15)$ |
| $ 6, 6, 3, 3 $ | $9$ | $6$ | $( 1, 4,17)( 2, 3,18)( 5,11, 7,13,10,15)( 6,12, 8,14, 9,16)$ |
| $ 6, 6, 3, 3 $ | $9$ | $6$ | $( 1, 4,17)( 2, 3,18)( 5,11,10,15, 7,13)( 6,12, 9,16, 8,14)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $9$ | $6$ | $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,18)(10,17)(11,16,13,12,15,14)$ |
| $ 6, 6, 6 $ | $9$ | $6$ | $( 1, 5, 4, 7,17,10)( 2, 6, 3, 8,18, 9)(11,16,13,12,15,14)$ |
| $ 6, 6, 6 $ | $18$ | $6$ | $( 1, 5,12, 2, 6,11)( 3, 8,13, 4, 7,14)( 9,15,17,10,16,18)$ |
| $ 18 $ | $18$ | $18$ | $( 1, 5,12, 3, 8,13,17,10,16, 2, 6,11, 4, 7,14,18, 9,15)$ |
| $ 18 $ | $18$ | $18$ | $( 1, 5,12,18, 9,15, 4, 7,14, 2, 6,11,17,10,16, 3, 8,13)$ |
| $ 6, 6, 6 $ | $9$ | $6$ | $( 1, 5,17,10, 4, 7)( 2, 6,18, 9, 3, 8)(11,16,13,12,15,14)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $9$ | $6$ | $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,17)(10,18)(11,15,13)(12,16,14)$ |
| $ 6, 6, 3, 3 $ | $9$ | $6$ | $( 1, 6, 4, 8,17, 9)( 2, 5, 3, 7,18,10)(11,15,13)(12,16,14)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1, 6,12)( 2, 5,11)( 3, 7,13)( 4, 8,14)( 9,16,17)(10,15,18)$ |
| $ 9, 9 $ | $18$ | $9$ | $( 1, 6,12, 4, 8,14,17, 9,16)( 2, 5,11, 3, 7,13,18,10,15)$ |
| $ 9, 9 $ | $18$ | $9$ | $( 1, 6,12,17, 9,16, 4, 8,14)( 2, 5,11,18,10,15, 3, 7,13)$ |
| $ 6, 6, 3, 3 $ | $9$ | $6$ | $( 1, 6,17, 9, 4, 8)( 2, 5,18,10, 3, 7)(11,15,13)(12,16,14)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,17, 4)( 2,18, 3)( 5,10, 7)( 6, 9, 8)(11,15,13)(12,16,14)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1,18, 4, 2,17, 3)( 5, 9, 7, 6,10, 8)(11,16,13,12,15,14)$ |
Group invariants
| Order: | $324=2^{2} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [324, 68] |
| Character table: Data not available. |