Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $203$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,16,4,18,6,13,2,15,3,17,5,14)(7,12,10,8,11,9), (1,12,5,10,3,8,2,11,6,9,4,7)(13,14)(15,16)(17,18) | |
| $|\Aut(F/K)|$: | $6$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $S_3$, $C_6$ 18: $S_3\times C_3$ 24: $S_4$ 54: $C_3^2 : C_6$ 72: 12T45 162: $C_3 \wr S_3 $ 216: 18T97 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: $S_4$
Degree 9: $C_3 \wr S_3 $
Low degree siblings
18T203 x 2, 18T208 x 3Siblings 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, 3, 3 $ | $1$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,11,10)( 8,12, 9)(13,17,16)(14,18,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,10,11)( 8, 9,12)(13,16,17)(14,15,18)$ |
| $ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,10,11)( 8, 9,12)(13,15,17,14,16,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)(13,14)(15,16)(17,18)$ |
| $ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,11,10)( 8,12, 9)(13,18,16,14,17,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $72$ | $3$ | $( 1,11,17)( 2,12,18)( 3, 7,13)( 4, 8,14)( 5, 9,15)( 6,10,16)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $6$ | $6$ | $( 7, 9,11, 8,10,12)(13,18,16,14,17,15)$ |
| $ 6, 3, 3, 2, 2, 2 $ | $6$ | $6$ | $( 1, 6, 3)( 2, 5, 4)( 7, 8)( 9,10)(11,12)(13,15,17,14,16,18)$ |
| $ 6, 3, 3, 2, 2, 2 $ | $6$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7,12,10, 8,11, 9)(13,14)(15,16)(17,18)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 7,10,11)( 8, 9,12)(13,17,16)(14,18,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 7,14)( 8,13)( 9,16)(10,15)(11,18)(12,17)$ |
| $ 6, 6, 3, 3 $ | $18$ | $6$ | $( 1, 6, 3)( 2, 5, 4)( 7,18,10,14,11,15)( 8,17, 9,13,12,16)$ |
| $ 6, 6, 3, 3 $ | $18$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7,15,11,14,10,18)( 8,16,12,13, 9,17)$ |
| $ 12, 6 $ | $18$ | $12$ | $( 1, 4, 6, 2, 3, 5)( 7,16,12,14,10,17, 8,15,11,13, 9,18)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $18$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7,13, 8,14)( 9,15,10,16)(11,17,12,18)$ |
| $ 12, 6 $ | $18$ | $12$ | $( 1, 5, 3, 2, 6, 4)( 7,17, 9,14,11,16, 8,18,10,13,12,15)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $(13,17,16)(14,18,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,11,10)( 8,12, 9)(13,16,17)(14,15,18)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,10,11)( 8, 9,12)$ |
| $ 6, 3, 3, 2, 2, 2 $ | $6$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,10,11)( 8, 9,12)(13,14)(15,16)(17,18)$ |
| $ 6, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)(13,18,16,14,17,15)$ |
| $ 6, 6, 3, 3 $ | $6$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,11,10)( 8,12, 9)(13,15,17,14,16,18)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $3$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7, 9,11, 8,10,12)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,17,16)(14,18,15)$ |
| $ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,12,10, 8,11, 9)(13,16,17)(14,15,18)$ |
| $ 9, 9 $ | $72$ | $9$ | $( 1,11,17, 6,10,16, 3, 7,13)( 2,12,18, 5, 9,15, 4, 8,14)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 7,14,11,18,10,15)( 8,13,12,17, 9,16)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $18$ | $6$ | $( 1, 6, 3)( 2, 5, 4)( 7,18)( 8,17)( 9,13)(10,14)(11,15)(12,16)$ |
| $ 6, 6, 3, 3 $ | $18$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7,15,10,18,11,14)( 8,16, 9,17,12,13)$ |
| $ 12, 6 $ | $18$ | $12$ | $( 1, 4, 6, 2, 3, 5)( 7,16, 9,18,11,13, 8,15,10,17,12,14)$ |
| $ 12, 2, 2, 2 $ | $18$ | $12$ | $( 1, 2)( 3, 4)( 5, 6)( 7,13,12,18,10,16, 8,14,11,17, 9,15)$ |
| $ 6, 4, 4, 4 $ | $18$ | $12$ | $( 1, 5, 3, 2, 6, 4)( 7,17, 8,18)( 9,14,10,13)(11,16,12,15)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $(13,16,17)(14,15,18)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,11,10)( 8,12, 9)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,10,11)( 8, 9,12)(13,17,16)(14,18,15)$ |
| $ 6, 6, 3, 3 $ | $6$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,10,11)( 8, 9,12)(13,18,16,14,17,15)$ |
| $ 6, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)(13,15,17,14,16,18)$ |
| $ 6, 3, 3, 2, 2, 2 $ | $6$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,11,10)( 8,12, 9)(13,14)(15,16)(17,18)$ |
| $ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7, 9,11, 8,10,12)(13,17,16)(14,18,15)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16,17)(14,15,18)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $3$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,12,10, 8,11, 9)$ |
| $ 9, 9 $ | $72$ | $9$ | $( 1,11,17, 3, 7,13, 6,10,16)( 2,12,18, 4, 8,14, 5, 9,15)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 7,14,10,15,11,18)( 8,13, 9,16,12,17)$ |
| $ 6, 6, 3, 3 $ | $18$ | $6$ | $( 1, 6, 3)( 2, 5, 4)( 7,18,11,15,10,14)( 8,17,12,16, 9,13)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $18$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7,15)( 8,16)( 9,17)(10,18)(11,14)(12,13)$ |
| $ 6, 4, 4, 4 $ | $18$ | $12$ | $( 1, 4, 6, 2, 3, 5)( 7,16, 8,15)( 9,18,10,17)(11,13,12,14)$ |
| $ 12, 2, 2, 2 $ | $18$ | $12$ | $( 1, 2)( 3, 4)( 5, 6)( 7,13, 9,15,11,17, 8,14,10,16,12,18)$ |
| $ 12, 6 $ | $18$ | $12$ | $( 1, 5, 3, 2, 6, 4)( 7,17,12,15,10,13, 8,18,11,16, 9,14)$ |
Group invariants
| Order: | $648=2^{3} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [648, 266] |
| Character table: Data not available. |