Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $301$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,15,4,13,17,11)(2,16,3,14,18,12)(5,7)(6,8), (1,10,16,18,8,11,4,5,14,2,9,15,17,7,12,3,6,13) | |
| $|\Aut(F/K)|$: | $2$ |
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: $S_4\times C_2$ 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: $S_4\times C_2$
Degree 9: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Low degree siblings
18T301, 18T313 x 2Siblings 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, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 1, 4,17)( 2, 3,18)(11,15,13)(12,16,14)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,17, 4)( 2,18, 3)( 5, 7,10)( 6, 8, 9)(11,15,13)(12,16,14)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $(11,13,15)(12,14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1,18)( 2,17)( 3, 4)(11,12)(13,16)(14,15)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $54$ | $6$ | $( 1,18)( 2,17)( 3, 4)( 5, 7,10)( 6, 8, 9)(11,14)(12,13)(15,16)$ |
| $ 9, 9 $ | $72$ | $9$ | $( 1,13, 7,17,11,10, 4,15, 5)( 2,14, 8,18,12, 9, 3,16, 6)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $72$ | $3$ | $( 1,15, 5)( 2,16, 6)( 3,12, 9)( 4,11,10)( 7,17,13)( 8,18,14)$ |
| $ 9, 9 $ | $72$ | $9$ | $( 1,11,10, 4,13, 7,17,15, 5)( 2,12, 9, 3,14, 8,18,16, 6)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $108$ | $6$ | $( 3,18)( 4,17)( 5,11, 7,13,10,15)( 6,12, 8,14, 9,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $54$ | $2$ | $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,11)( 8,12)( 9,14)(10,13)$ |
| $ 6, 4, 4, 4 $ | $54$ | $12$ | $( 1,18, 4, 2,17, 3)( 5,12, 8,15)( 6,11, 7,16)( 9,13,10,14)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $54$ | $4$ | $( 1, 2)( 3, 4)( 5,16, 6,15)( 7,14, 9,11)( 8,13,10,12)(17,18)$ |
| $ 6, 4, 4, 4 $ | $54$ | $12$ | $( 1, 3,17, 2, 4,18)( 5,14, 9,15)( 6,13,10,16)( 7,12, 8,11)$ |
| $ 6, 2, 2, 2, 2, 1, 1, 1, 1 $ | $54$ | $6$ | $( 3,18)( 4,17)( 7,10)( 8, 9)(11,14,15,12,13,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 4)( 2, 3)( 7,10)( 8, 9)(11,12)(13,14)(15,16)$ |
| $ 6, 6, 2, 2, 2 $ | $12$ | $6$ | $( 1,18, 4, 2,17, 3)( 5, 6)( 7, 8)( 9,10)(11,14,15,12,13,16)$ |
| $ 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, 6, 6 $ | $8$ | $6$ | $( 1,18, 4, 2,17, 3)( 5, 8,10, 6, 7, 9)(11,16,13,12,15,14)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $6$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,14,15,12,13,16)(17,18)$ |
| $ 6, 6, 6 $ | $72$ | $6$ | $( 1,13, 6, 2,14, 5)( 3,12, 7, 4,11, 8)( 9,18,16,10,17,15)$ |
| $ 18 $ | $72$ | $18$ | $( 1,11, 8,18,14, 5, 4,15, 9, 2,12, 7,17,13, 6, 3,16,10)$ |
| $ 18 $ | $72$ | $18$ | $( 1,15, 9, 3,14, 5,17,11, 8, 2,16,10, 4,13, 6,18,12, 7)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $54$ | $4$ | $( 5,11, 9,14)( 6,12,10,13)( 7,15, 8,16)$ |
| $ 4, 4, 4, 3, 3 $ | $54$ | $12$ | $( 1, 4,17)( 2, 3,18)( 5,15, 8,14)( 6,16, 7,13)( 9,12,10,11)$ |
| $ 4, 4, 4, 3, 3 $ | $54$ | $12$ | $( 1,17, 4)( 2,18, 3)( 5,13, 6,14)( 7,11, 9,16)( 8,12,10,15)$ |
| $ 6, 6, 2, 2, 2 $ | $108$ | $6$ | $( 1,18)( 2,17)( 3, 4)( 5,12,10,16, 7,14)( 6,11, 9,15, 8,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $54$ | $2$ | $( 1, 3)( 2, 4)( 5,14)( 6,13)( 7,16)( 8,15)( 9,11)(10,12)(17,18)$ |
Group invariants
| Order: | $1296=2^{4} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [1296, 3492] |
| Character table: Data not available. |