Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $718$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,3,5)(2,4,6)(7,18)(8,17)(9,15,10,16)(11,14,12,13), (1,16,2,15)(3,13,5,17,4,14,6,18)(9,12)(10,11) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 6: $S_3$ 24: $S_4$ 648: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: None
Degree 9: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Low degree siblings
12T290, 18T712, 18T716, 18T720Siblings 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 $ | $9$ | $2$ | $(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 7, 8)(11,12)(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $3$ | $( 1, 3, 5)( 2, 4, 6)$ |
| $ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $144$ | $6$ | $( 1, 3, 5)( 2, 4, 6)(13,14)(17,18)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $216$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7, 8)(11,12)(13,14)(17,18)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $192$ | $3$ | $( 1, 3, 5)( 2, 4, 6)(13,17,16)(14,18,15)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $576$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7, 8)(11,12)(13,17,16)(14,18,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $256$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,17,16)(14,18,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $256$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,10,12)( 8, 9,11)(13,17,16)(14,18,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $108$ | $2$ | $( 1, 5)( 2, 6)( 9,11)(10,12)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $324$ | $2$ | $( 1, 5)( 2, 6)( 9,11)(10,12)(13,14)(17,18)$ |
| $ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $216$ | $4$ | $( 1, 5)( 2, 6)( 7, 8)( 9,12,10,11)$ |
| $ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $648$ | $4$ | $( 1, 5)( 2, 6)( 7, 8)( 9,12,10,11)(13,14)(17,18)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $108$ | $4$ | $( 1, 6, 2, 5)( 3, 4)( 7, 8)( 9,12,10,11)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $324$ | $4$ | $( 1, 6, 2, 5)( 3, 4)( 7, 8)( 9,12,10,11)(13,14)(17,18)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $864$ | $6$ | $( 1, 5)( 2, 6)( 9,11)(10,12)(13,17,16)(14,18,15)$ |
| $ 4, 3, 3, 2, 2, 2, 1, 1 $ | $1728$ | $12$ | $( 1, 5)( 2, 6)( 7, 8)( 9,12,10,11)(13,17,16)(14,18,15)$ |
| $ 4, 4, 3, 3, 2, 2 $ | $864$ | $12$ | $( 1, 6, 2, 5)( 3, 4)( 7, 8)( 9,12,10,11)(13,17,16)(14,18,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1152$ | $3$ | $( 1,14, 7)( 2,13, 8)( 3,18, 9)( 4,17,10)( 5,16,12)( 6,15,11)$ |
| $ 6, 6, 3, 3 $ | $3456$ | $6$ | $( 1,13, 8, 2,14, 7)( 3,17,10, 4,18, 9)( 5,16,12)( 6,15,11)$ |
| $ 9, 9 $ | $4608$ | $9$ | $( 1,14, 7, 3,18, 9, 5,16,12)( 2,13, 8, 4,17,10, 6,15,11)$ |
| $ 9, 9 $ | $4608$ | $9$ | $( 1,14, 7, 5,16,12, 3,18, 9)( 2,13, 8, 6,15,11, 4,17,10)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $1728$ | $6$ | $( 1, 9, 6, 8, 4,11)( 2,10, 5, 7, 3,12)(15,16)(17,18)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $576$ | $6$ | $( 1, 9, 6, 8, 4,11)( 2,10, 5, 7, 3,12)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $648$ | $4$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5, 7)( 6, 8)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $216$ | $2$ | $( 1,10)( 2, 9)( 3,11)( 4,12)( 5, 7)( 6, 8)(15,16)(17,18)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $216$ | $4$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5, 7)( 6, 8)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $72$ | $2$ | $( 1,10)( 2, 9)( 3,11)( 4,12)( 5, 7)( 6, 8)$ |
| $ 6, 6, 3, 3 $ | $2304$ | $6$ | $( 1, 9, 6, 8, 4,11)( 2,10, 5, 7, 3,12)(13,17,15)(14,18,16)$ |
| $ 4, 4, 3, 3, 2, 2 $ | $864$ | $12$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5, 7)( 6, 8)(13,17,15)(14,18,16)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $288$ | $6$ | $( 1,10)( 2, 9)( 3,11)( 4,12)( 5, 7)( 6, 8)(13,17,15)(14,18,16)$ |
| $ 6, 6, 3, 3 $ | $2304$ | $6$ | $( 1, 9, 6, 8, 4,11)( 2,10, 5, 7, 3,12)(13,16,18)(14,15,17)$ |
| $ 4, 4, 3, 3, 2, 2 $ | $864$ | $12$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5, 7)( 6, 8)(13,16,18)(14,15,17)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $288$ | $6$ | $( 1,10)( 2, 9)( 3,11)( 4,12)( 5, 7)( 6, 8)(13,16,18)(14,15,17)$ |
| $ 8, 4, 2, 2, 1, 1 $ | $2592$ | $8$ | $( 1, 9, 6, 8, 2,10, 5, 7)( 3,12, 4,11)(15,17)(16,18)$ |
| $ 8, 4, 4, 2 $ | $2592$ | $8$ | $( 1, 9, 6, 8, 2,10, 5, 7)( 3,12, 4,11)(13,14)(15,18,16,17)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $2592$ | $4$ | $( 1, 9, 6, 7)( 2,10, 5, 8)( 3,11)( 4,12)(15,17)(16,18)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $2592$ | $4$ | $( 1, 9, 6, 7)( 2,10, 5, 8)( 3,11)( 4,12)(13,14)(15,18,16,17)$ |
Group invariants
| Order: | $41472=2^{9} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |