Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $228$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,8,3,12,6,10,2,7,4,11,5,9)(13,14)(15,17)(16,18), (1,2)(3,4)(5,6)(7,10,12,8,9,11)(13,17,15,14,18,16), (1,12,18,5,8,16)(2,11,17,6,7,15)(3,10,14,4,9,13) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_2^2$ x 7 6: $S_3$ x 2 8: $C_2^3$ 12: $D_{6}$ x 6 24: $S_4$, $S_3 \times C_2^2$ x 2 36: $S_3^2$ 48: $S_4\times C_2$ x 3 72: 12T37 96: 12T48 108: $C_3^2 : D_{6} $ 144: 12T83 216: 18T94 288: 18T111 432: 18T152 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: $S_4\times C_2$
Degree 9: $C_3^2 : D_{6} $
Low degree siblings
18T228 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, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 1, 6, 4)( 2, 5, 3)( 7, 9,12)( 8,10,11)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 6, 4)( 2, 5, 3)( 7,12, 9)( 8,11,10)(13,18,15)(14,17,16)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $6$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7,11, 9, 8,12,10)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| $ 6, 3, 3, 2, 2, 2 $ | $12$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7, 8)( 9,10)(11,12)(13,18,15)(14,17,16)$ |
| $ 6, 6, 3, 3 $ | $6$ | $6$ | $( 1, 5, 4, 2, 6, 3)( 7,11, 9, 8,12,10)(13,18,15)(14,17,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $24$ | $3$ | $( 1,17, 8)( 2,18, 7)( 3,13, 9)( 4,14,10)( 5,15,12)( 6,16,11)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $48$ | $3$ | $( 1,17,10)( 2,18, 9)( 3,13,12)( 4,14,11)( 5,15, 7)( 6,16, 8)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $36$ | $6$ | $( 3, 5)( 4, 6)( 7,14, 9,17,12,16)( 8,13,10,18,11,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $18$ | $2$ | $( 1, 4)( 2, 3)( 7,14)( 8,13)( 9,17)(10,18)(11,15)(12,16)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $18$ | $4$ | $( 1, 3)( 2, 4)( 5, 6)( 7,14, 8,13)( 9,17,10,18)(11,15,12,16)$ |
| $ 12, 2, 2, 2 $ | $36$ | $12$ | $( 1, 2)( 3, 6)( 4, 5)( 7,14,10,18,12,16, 8,13, 9,17,11,15)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $(13,14)(15,16)(17,18)$ |
| $ 3, 3, 3, 3, 2, 2, 2 $ | $6$ | $6$ | $( 1, 6, 4)( 2, 5, 3)( 7, 9,12)( 8,10,11)(13,14)(15,16)(17,18)$ |
| $ 6, 3, 3, 1, 1, 1, 1, 1, 1 $ | $12$ | $6$ | $( 7, 9,12)( 8,10,11)(13,17,15,14,18,16)$ |
| $ 6, 3, 3, 3, 3 $ | $6$ | $6$ | $( 1, 6, 4)( 2, 5, 3)( 7,12, 9)( 8,11,10)(13,17,15,14,18,16)$ |
| $ 6, 6, 2, 2, 2 $ | $6$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7,11, 9, 8,12,10)(13,14)(15,16)(17,18)$ |
| $ 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 $ | $2$ | $6$ | $( 1, 5, 4, 2, 6, 3)( 7,11, 9, 8,12,10)(13,17,15,14,18,16)$ |
| $ 6, 6, 6 $ | $24$ | $6$ | $( 1,17, 7, 2,18, 8)( 3,13,10, 4,14, 9)( 5,15,11, 6,16,12)$ |
| $ 6, 6, 6 $ | $48$ | $6$ | $( 1,17, 9, 2,18,10)( 3,13,11, 4,14,12)( 5,15, 8, 6,16, 7)$ |
| $ 12, 2, 2, 1, 1 $ | $36$ | $12$ | $( 3, 5)( 4, 6)( 7,14,10,18,12,16, 8,13, 9,17,11,15)$ |
| $ 4, 4, 4, 2, 2, 1, 1 $ | $18$ | $4$ | $( 1, 4)( 2, 3)( 7,14, 8,13)( 9,17,10,18)(11,15,12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $18$ | $2$ | $( 1, 3)( 2, 4)( 5, 6)( 7,14)( 8,13)( 9,17)(10,18)(11,15)(12,16)$ |
| $ 6, 6, 2, 2, 2 $ | $36$ | $6$ | $( 1, 2)( 3, 6)( 4, 5)( 7,14, 9,17,12,16)( 8,13,10,18,11,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3, 5)( 4, 6)( 9,12)(10,11)(13,16)(14,15)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 3)( 2, 4)( 5, 6)( 7,11)( 8,12)( 9,10)(13,16)(14,15)(17,18)$ |
| $ 6, 6, 6 $ | $72$ | $6$ | $( 1,17, 7, 2,18, 8)( 3,15,10, 6,14,12)( 4,16, 9, 5,13,11)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $18$ | $4$ | $( 7,14, 8,13)( 9,16,10,15)(11,18,12,17)$ |
| $ 12, 3, 3 $ | $36$ | $12$ | $( 1, 6, 4)( 2, 5, 3)( 7,14,10,15,12,17, 8,13, 9,16,11,18)$ |
| $ 6, 6, 6 $ | $36$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7,14,12,17, 9,16)( 8,13,11,18,10,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $18$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,14)( 8,13)( 9,16)(10,15)(11,18)(12,17)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 5)( 4, 6)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $27$ | $2$ | $( 1, 3)( 2, 4)( 5, 6)( 7,11)( 8,12)( 9,10)(13,15)(14,16)$ |
| $ 6, 6, 3, 3 $ | $72$ | $6$ | $( 1,17, 8)( 2,18, 7)( 3,15, 9, 5,13,12)( 4,16,10, 6,14,11)$ |
| $ 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 $ | $36$ | $6$ | $( 1, 6, 4)( 2, 5, 3)( 7,14, 9,16,12,17)( 8,13,10,15,11,18)$ |
| $ 12, 6 $ | $36$ | $12$ | $( 1, 3, 6, 2, 4, 5)( 7,14,11,18, 9,16, 8,13,12,17,10,15)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $18$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7,14, 8,13)( 9,16,10,15)(11,18,12,17)$ |
Group invariants
| Order: | $864=2^{5} \cdot 3^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [864, 4000] |
| Character table: Data not available. |