Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $566$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,17,3,14,5,16)(2,18,4,13,6,15), (1,8,2,7)(3,9)(4,10)(5,11,6,12)(13,14)(17,18) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $S_3$, $C_6$ 18: $S_3\times C_3$ 54: $C_3^2 : C_6$ 162: $C_3 \wr S_3 $ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: None
Degree 9: $C_3 \wr S_3 $
Low degree siblings
12T275, 18T564Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 5, 6)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $48$ | $3$ | $( 1, 3, 5)( 2, 4, 6)(13,15,18)(14,16,17)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $144$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7, 8)(11,12)(13,15,18)(14,16,17)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $48$ | $3$ | $( 1, 5, 3)( 2, 6, 4)(13,18,15)(14,17,16)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $144$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7, 8)(11,12)(13,18,15)(14,17,16)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $96$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,10,12)( 8, 9,11)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $288$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7,10,12)( 8, 9,11)(13,14)(15,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $192$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,18,15)(14,17,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $192$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,12,10)( 8,11, 9)(13,15,18)(14,16,17)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 7,12,10)( 8,11, 9)$ |
| $ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $72$ | $6$ | $( 7,12,10)( 8,11, 9)(13,14)(15,16)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $108$ | $6$ | $( 1, 2)( 5, 6)( 7,12,10)( 8,11, 9)(13,14)(15,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,12,10)( 8,11, 9)(13,18,15)(14,17,16)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $(13,15,18)(14,16,17)$ |
| $ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $72$ | $6$ | $( 1, 2)( 5, 6)(13,15,18)(14,16,17)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $108$ | $6$ | $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,15,18)(14,16,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,15,18)(14,16,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $288$ | $3$ | $( 1, 8,16)( 2, 7,15)( 3,10,17)( 4, 9,18)( 5,12,14)( 6,11,13)$ |
| $ 6, 6, 3, 3 $ | $864$ | $6$ | $( 1, 8,15, 2, 7,16)( 3,10,17)( 4, 9,18)( 5,12,13, 6,11,14)$ |
| $ 9, 9 $ | $1152$ | $9$ | $( 1, 8,17, 5,12,16, 3,10,14)( 2, 7,18, 6,11,15, 4, 9,13)$ |
| $ 9, 9 $ | $1152$ | $9$ | $( 1, 8,14, 3,10,16, 5,12,17)( 2, 7,13, 4, 9,15, 6,11,18)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $144$ | $6$ | $( 1,10, 3,12, 5, 8)( 2, 9, 4,11, 6, 7)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $432$ | $6$ | $( 1,10, 3,12, 5, 8)( 2, 9, 4,11, 6, 7)(13,14)(15,16)$ |
| $ 6, 6, 3, 3 $ | $576$ | $6$ | $( 1,10, 5, 8, 3,12)( 2, 9, 6, 7, 4,11)(13,15,18)(14,16,17)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $144$ | $6$ | $( 1,10)( 2, 9)( 3,12)( 4,11)( 5, 8)( 6, 7)(13,18,15)(14,17,16)$ |
| $ 4, 4, 3, 3, 2, 2 $ | $432$ | $12$ | $( 1,10, 2, 9)( 3,12)( 4,11)( 5, 8, 6, 7)(13,18,15)(14,17,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $108$ | $4$ | $( 1, 7, 2, 8)( 3,10)( 4, 9)( 5,11, 6,12)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $324$ | $4$ | $( 1, 7, 2, 8)( 3,10)( 4, 9)( 5,11, 6,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $2$ | $( 1, 7)( 2, 8)( 3,10)( 4, 9)( 5,11)( 6,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $108$ | $2$ | $( 1, 7)( 2, 8)( 3,10)( 4, 9)( 5,11)( 6,12)(13,14)(15,16)$ |
| $ 6, 6, 3, 3 $ | $576$ | $6$ | $( 1, 7, 4, 9, 6,12)( 2, 8, 3,10, 5,11)(13,15,18)(14,16,17)$ |
| $ 6, 6, 3, 3 $ | $576$ | $6$ | $( 1, 7, 6,12, 3,10)( 2, 8, 5,11, 4, 9)(13,18,15)(14,17,16)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $144$ | $6$ | $( 1,12, 5, 9, 4, 8)( 2,11, 6,10, 3, 7)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $432$ | $6$ | $( 1,12, 5, 9, 4, 8)( 2,11, 6,10, 3, 7)(13,14)(15,16)$ |
| $ 4, 4, 3, 3, 2, 2 $ | $432$ | $12$ | $( 1,12)( 2,11)( 3, 7, 4, 8)( 5, 9, 6,10)(13,15,18)(14,16,17)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $144$ | $6$ | $( 1,11)( 2,12)( 3, 8)( 4, 7)( 5, 9)( 6,10)(13,15,18)(14,16,17)$ |
| $ 6, 6, 3, 3 $ | $576$ | $6$ | $( 1,12, 3, 7, 6,10)( 2,11, 4, 8, 5, 9)(13,18,15)(14,17,16)$ |
Group invariants
| Order: | $10368=2^{7} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |