Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $319$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,8,13)(2,7,14)(3,10,15)(4,9,16)(5,12,18)(6,11,17), (1,3,17)(2,4,18), (1,3)(2,4), (1,2)(3,4)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(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: $S_3$ 12: $D_{6}$ 24: $S_4$ 48: $S_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $S_3$
Degree 9: $S_3\wr S_3$
Low degree siblings
9T31, 12T213, 18T300, 18T303, 18T311, 18T312, 18T314, 18T315, 18T320Siblings 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, 3, 3 $ | $8$ | $3$ | $( 1,17, 3)( 2,18, 4)( 5, 9, 7)( 6,10, 8)(11,15,13)(12,16,14)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 5, 7, 9)( 6, 8,10)(11,13,15)(12,14,16)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 1,17, 3)( 2,18, 4)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3,17)( 4,18)(13,15)(14,16)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $54$ | $6$ | $( 1,17)( 2,18)( 5, 9, 7)( 6,10, 8)(11,15)(12,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $72$ | $3$ | $( 1, 8,13)( 2, 7,14)( 3,10,15)( 4, 9,16)( 5,12,18)( 6,11,17)$ |
| $ 9, 9 $ | $144$ | $9$ | $( 1,10,11,17, 8,15, 3, 6,13)( 2, 9,12,18, 7,16, 4, 5,14)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(13,15)(14,16)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1,17, 3)( 2,18, 4)( 5, 9, 7)( 6,10, 8)(11,15)(12,16)$ |
| $ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $6$ | $( 5, 7, 9)( 6, 8,10)(11,13)(12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3,17)( 4,18)( 7, 9)( 8,10)(13,15)(14,16)$ |
| $ 6, 6, 3, 3 $ | $216$ | $6$ | $( 1, 8,13, 3,10,15)( 2, 7,14, 4, 9,16)( 5,12,18)( 6,11,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $18$ | $2$ | $( 1, 2)( 3, 4)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)(17,18)$ |
| $ 6, 6, 6 $ | $72$ | $6$ | $( 1,18, 3, 2,17, 4)( 5,15, 7,11, 9,13)( 6,16, 8,12,10,14)$ |
| $ 6, 6, 2, 2, 2 $ | $36$ | $6$ | $( 1, 2)( 3, 4)( 5,13, 9,11, 7,15)( 6,14,10,12, 8,16)(17,18)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $36$ | $6$ | $( 1,18, 3, 2,17, 4)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $162$ | $4$ | $( 1, 2)( 3,18)( 4,17)( 5,11)( 6,12)( 7,15, 9,13)( 8,16,10,14)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $54$ | $4$ | $( 1, 2)( 3, 4)( 5,11)( 6,12)( 7,15, 9,13)( 8,16,10,14)(17,18)$ |
| $ 6, 4, 4, 2, 2 $ | $108$ | $12$ | $( 1,18, 3, 2,17, 4)( 5,15, 7,13)( 6,16, 8,14)( 9,11)(10,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $54$ | $2$ | $( 1, 2)( 3,18)( 4,17)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)$ |
| $ 6, 6, 2, 2, 2 $ | $108$ | $6$ | $( 1,18)( 2,17)( 3, 4)( 5,15, 7,11, 9,13)( 6,16, 8,12,10,14)$ |
Group invariants
| Order: | $1296=2^{4} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [1296, 3490] |
| Character table: Data not available. |