Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $521$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,13,9)(2,14,10)(3,16,11)(4,15,12)(5,18,8,6,17,7), (1,15,10)(2,16,9)(3,18,12,4,17,11)(5,13,8,6,14,7), (1,12)(2,11)(3,7,4,8)(5,9,6,10)(13,14)(17,18) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 6: $S_3$ x 4 12: $D_{6}$ x 4 18: $C_3^2:C_2$ 36: 18T12 54: $(C_3^2:C_3):C_2$ 108: 18T52 3456: 12T252 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
18T521 x 7Siblings 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, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)( 9,10)(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 5, 6)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 7, 8)( 9,10)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $96$ | $3$ | $( 1, 9,13)( 2,10,14)( 3,11,16)( 4,12,15)( 5, 8,17)( 6, 7,18)$ |
| $ 6, 6, 3, 3 $ | $288$ | $6$ | $( 1,10,13, 2, 9,14)( 3,11,16)( 4,12,15)( 5, 7,17, 6, 8,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,11, 9)( 8,12,10)(13,18,16)(14,17,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,16,18)(14,15,17)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5, 6)(11,12)(17,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 7, 8)( 9,10)(11,12)(13,14)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 4)( 9,10)(13,14)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5, 6)( 9,10)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(15,16)$ |
| $ 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, 3, 3, 3, 3 $ | $288$ | $6$ | $( 1, 9,13)( 2,10,14)( 3,11,15, 4,12,16)( 5, 7,18)( 6, 8,17)$ |
| $ 6, 6, 6 $ | $96$ | $6$ | $( 1,10,13, 2, 9,14)( 3,11,15, 4,12,16)( 5, 8,18, 6, 7,17)$ |
| $ 6, 6, 6 $ | $64$ | $6$ | $( 1, 6, 4, 2, 5, 3)( 7,11,10, 8,12, 9)(13,18,15,14,17,16)$ |
| $ 6, 6, 6 $ | $64$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7, 9,11, 8,10,12)(13,16,18,14,15,17)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $96$ | $3$ | $( 7, 9,12)( 8,10,11)(13,18,15)(14,17,16)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $288$ | $6$ | $( 1, 2)( 5, 6)( 7,10,11)( 8, 9,12)(13,17,16)(14,18,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $96$ | $3$ | $( 1, 9,15)( 2,10,16)( 3,11,17)( 4,12,18)( 5, 8,14)( 6, 7,13)$ |
| $ 6, 6, 3, 3 $ | $288$ | $6$ | $( 1,10,16)( 2, 9,15)( 3,11,18, 4,12,17)( 5, 7,14, 6, 8,13)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $96$ | $3$ | $( 1,13, 7)( 2,14, 8)( 3,16,10)( 4,15, 9)( 5,17,11)( 6,18,12)$ |
| $ 6, 6, 3, 3 $ | $288$ | $6$ | $( 1,14, 7, 2,13, 8)( 3,16, 9, 4,15,10)( 5,18,12)( 6,17,11)$ |
| $ 6, 6, 2, 1, 1, 1, 1 $ | $288$ | $6$ | $( 5, 6)( 7, 9,11, 8,10,12)(13,17,16,14,18,15)$ |
| $ 6, 6, 2, 2, 2 $ | $96$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 9,12, 8,10,11)(13,17,16,14,18,15)$ |
| $ 6, 3, 3, 3, 3 $ | $288$ | $6$ | $( 1, 9,16, 2,10,15)( 3,11,17)( 4,12,18)( 5, 7,13)( 6, 8,14)$ |
| $ 6, 6, 6 $ | $96$ | $6$ | $( 1,10,15, 2, 9,16)( 3,11,18, 4,12,17)( 5, 8,13, 6, 7,14)$ |
| $ 6, 6, 6 $ | $96$ | $6$ | $( 1,13, 8, 2,14, 7)( 3,16,10, 4,15, 9)( 5,18,12, 6,17,11)$ |
| $ 6, 3, 3, 3, 3 $ | $288$ | $6$ | $( 1,14, 8)( 2,13, 7)( 3,16, 9)( 4,15,10)( 5,17,11, 6,18,12)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $108$ | $4$ | $( 7,13)( 8,14)( 9,16,10,15)(11,18,12,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $108$ | $2$ | $( 1, 2)( 5, 6)( 7,14)( 8,13)( 9,16)(10,15)(11,17)(12,18)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $108$ | $4$ | $( 3, 4)( 5, 6)( 7,14, 8,13)( 9,16)(10,15)(11,17,12,18)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $108$ | $4$ | $( 1, 2)( 3, 4)( 7,13, 8,14)( 9,16,10,15)(11,18)(12,17)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $2$ | $( 7,14)( 8,13)( 9,15)(10,16)(11,18)(12,17)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $108$ | $4$ | $( 1, 2)( 5, 6)( 7,13)( 8,14)( 9,15,10,16)(11,17,12,18)$ |
| $ 6, 6, 3, 3 $ | $576$ | $6$ | $( 1, 9, 3,11, 6, 7)( 2,10, 4,12, 5, 8)(13,18,15)(14,17,16)$ |
| $ 6, 6, 3, 3 $ | $576$ | $6$ | $( 1,13, 6,18, 4,15)( 2,14, 5,17, 3,16)( 7, 9,11)( 8,10,12)$ |
| $ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ | $108$ | $4$ | $( 5, 6)( 7,13)( 8,14)( 9,16,10,15)(11,17,12,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $108$ | $2$ | $( 1, 2)( 7,14)( 8,13)( 9,16)(10,15)(11,18)(12,17)$ |
| $ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ | $108$ | $4$ | $( 3, 4)( 7,14, 8,13)( 9,16)(10,15)(11,18,12,17)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $108$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7,13, 8,14)( 9,16,10,15)(11,17)(12,18)$ |
| $ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ | $108$ | $4$ | $( 5, 6)( 7,13, 8,14)( 9,15)(10,16)(11,18,12,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $36$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,13)( 8,14)( 9,15)(10,16)(11,18)(12,17)$ |
| $ 6, 6, 6 $ | $576$ | $6$ | $( 1, 9, 3,11, 5, 7)( 2,10, 4,12, 6, 8)(13,18,16,14,17,15)$ |
| $ 6, 6, 6 $ | $576$ | $6$ | $( 1,13, 6,17, 4,15)( 2,14, 5,18, 3,16)( 7, 9,11, 8,10,12)$ |
Group invariants
| Order: | $6912=2^{8} \cdot 3^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |