Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $437$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,7,16)(2,8,15)(3,9,18,4,10,17)(5,12,13,6,11,14), (1,16,2,15)(3,17,4,18)(5,14)(6,13)(7,8)(11,12), (7,12,10)(8,11,9)(13,15,17)(14,16,18) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 6: $S_3$ x 4 18: $C_3^2:C_2$ 54: $(C_3^2:C_3):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: None
Degree 9: $(C_3^2:C_3):C_2$
Low degree siblings
12T252 x 4, 18T436 x 4, 18T437 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)( 9,10)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 4)( 5, 6)( 7, 8)(11,12)(15,16)(17,18)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(13,14)(17,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $96$ | $3$ | $( 1,16, 7)( 2,15, 8)( 3,18,10)( 4,17, 9)( 5,13,11)( 6,14,12)$ |
| $ 6, 6, 3, 3 $ | $288$ | $6$ | $( 1,15, 8, 2,16, 7)( 3,18, 9, 4,17,10)( 5,14,11)( 6,13,12)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,11,10)( 8,12, 9)(13,18,15)(14,17,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 4, 6)( 2, 3, 5)( 7,10,12)( 8, 9,11)(13,16,18)(14,15,17)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $96$ | $3$ | $( 7,10,12)( 8, 9,11)(13,17,15)(14,18,16)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $288$ | $6$ | $( 1, 2)( 3, 4)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $96$ | $3$ | $( 1,16,12)( 2,15,11)( 3,18, 7)( 4,17, 8)( 5,13, 9)( 6,14,10)$ |
| $ 6, 6, 3, 3 $ | $288$ | $6$ | $( 1,16,12, 2,15,11)( 3,18, 8)( 4,17, 7)( 5,13,10, 6,14, 9)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $96$ | $3$ | $( 1, 7,18)( 2, 8,17)( 3,10,14)( 4, 9,13)( 5,11,15)( 6,12,16)$ |
| $ 6, 6, 3, 3 $ | $288$ | $6$ | $( 1, 8,18)( 2, 7,17)( 3, 9,13, 4,10,14)( 5,11,16, 6,12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $108$ | $2$ | $( 3, 4)( 5, 6)( 7,15)( 8,16)( 9,17)(10,18)(11,14)(12,13)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $108$ | $4$ | $( 1, 2)( 5, 6)( 7,15, 8,16)( 9,17,10,18)(11,14)(12,13)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $108$ | $4$ | $( 3, 4)( 5, 6)( 7,16, 8,15)( 9,17,10,18)(11,13)(12,14)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $108$ | $4$ | $( 7,16, 8,15)( 9,17)(10,18)(11,13,12,14)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $108$ | $4$ | $( 3, 4)( 5, 6)( 7,15)( 8,16)( 9,18,10,17)(11,13,12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $2$ | $( 7,15)( 8,16)( 9,18)(10,17)(11,13)(12,14)$ |
| $ 6, 6, 3, 3 $ | $576$ | $6$ | $( 1,12, 3, 8, 6, 9)( 2,11, 4, 7, 5,10)(13,17,15)(14,18,16)$ |
| $ 6, 6, 3, 3 $ | $576$ | $6$ | $( 1,17, 5,16, 4,13)( 2,18, 6,15, 3,14)( 7, 9,12)( 8,10,11)$ |
Group invariants
| Order: | $3456=2^{7} \cdot 3^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |