Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $375$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,18,2,17)(3,15,4,16)(5,13)(6,14)(7,11)(8,12), (1,11)(2,12)(3,9,4,10)(5,8,6,7)(13,15,14,16) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 6: $S_3$ 12: $D_{6}$ 18: $D_{9}$ 36: $D_{18}$ 1152: 12T207 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: None
Degree 9: $D_{9}$
Low degree siblings
18T375 x 13Siblings 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 $ | $9$ | $2$ | $( 1, 2)( 3, 4)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 7, 8)( 9,10)(13,14)(17,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $128$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,10,12)( 8, 9,11)(13,16,17)(14,15,18)$ |
| $ 9, 9 $ | $128$ | $9$ | $( 1, 9,18, 3,11,14, 6, 8,15)( 2,10,17, 4,12,13, 5, 7,16)$ |
| $ 9, 9 $ | $128$ | $9$ | $( 1,11,15, 3, 8,18, 6, 9,14)( 2,12,16, 4, 7,17, 5,10,13)$ |
| $ 9, 9 $ | $128$ | $9$ | $( 1, 8,14, 3, 9,15, 6,11,18)( 2, 7,13, 4,10,16, 5,12,17)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $72$ | $4$ | $( 3, 6)( 4, 5)( 7,18)( 8,17)( 9,15,10,16)(11,13,12,14)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $72$ | $4$ | $( 1, 2)( 3, 6, 4, 5)( 7,17, 8,18)( 9,16)(10,15)(11,13,12,14)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $72$ | $4$ | $( 3, 6)( 4, 5)( 7,17, 8,18)( 9,15)(10,16)(11,14,12,13)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $72$ | $4$ | $( 1, 2)( 3, 6, 4, 5)( 7,18)( 8,17)( 9,16,10,15)(11,14,12,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $72$ | $2$ | $( 3, 6)( 4, 5)( 7,18)( 8,17)( 9,16)(10,15)(11,14)(12,13)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $72$ | $4$ | $( 1, 2)( 3, 6, 4, 5)( 7,17, 8,18)( 9,15,10,16)(11,14)(12,13)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $72$ | $4$ | $( 3, 6)( 4, 5)( 7,17, 8,18)( 9,16,10,15)(11,13)(12,14)$ |
| $ 4, 2, 2, 2, 2, 2, 2, 2 $ | $72$ | $4$ | $( 1, 2)( 3, 6, 4, 5)( 7,18)( 8,17)( 9,15)(10,16)(11,13)(12,14)$ |
| $ 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 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)(11,12)(15,16)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5, 6)( 9,10)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 9,10)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 7, 8)( 9,10)(11,12)(17,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)(11,12)(13,14)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 7, 8)(15,16)$ |
| $ 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, 6, 6 $ | $128$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7,10,12, 8, 9,11)(13,16,17,14,15,18)$ |
| $ 18 $ | $128$ | $18$ | $( 1, 9,18, 4,12,14, 6, 7,16, 2,10,17, 3,11,13, 5, 8,15)$ |
| $ 18 $ | $128$ | $18$ | $( 1,11,16, 4, 7,17, 6,10,13, 2,12,15, 3, 8,18, 5, 9,14)$ |
| $ 18 $ | $128$ | $18$ | $( 1, 8,14, 3, 9,15, 6,12,18, 2, 7,13, 4,10,16, 5,11,17)$ |
| $ 4, 4, 4, 2, 2, 1, 1 $ | $72$ | $4$ | $( 3, 6, 4, 5)( 7,18, 8,17)( 9,15,10,16)(11,14)(12,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $72$ | $2$ | $( 1, 2)( 3, 6)( 4, 5)( 7,17)( 8,18)( 9,16)(10,15)(11,14)(12,13)$ |
| $ 4, 2, 2, 2, 2, 2, 2, 1, 1 $ | $72$ | $4$ | $( 3, 6, 4, 5)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $72$ | $4$ | $( 1, 2)( 3, 6)( 4, 5)( 7,18, 8,17)( 9,16,10,15)(11,13)(12,14)$ |
| $ 4, 4, 4, 2, 2, 1, 1 $ | $72$ | $4$ | $( 3, 6, 4, 5)( 7,18, 8,17)( 9,16)(10,15)(11,13,12,14)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $72$ | $4$ | $( 1, 2)( 3, 6)( 4, 5)( 7,17)( 8,18)( 9,15,10,16)(11,13,12,14)$ |
| $ 4, 4, 4, 2, 2, 1, 1 $ | $72$ | $4$ | $( 3, 6, 4, 5)( 7,17)( 8,18)( 9,16,10,15)(11,14,12,13)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $72$ | $4$ | $( 1, 2)( 3, 6)( 4, 5)( 7,18, 8,17)( 9,15)(10,16)(11,14,12,13)$ |
Group invariants
| Order: | $2304=2^{8} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |