Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $368$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,18,15,14,11,9,7,5,3)(2,17,16,13,12,10,8,6,4), (1,12,4,13,5,15,7,17,10)(2,11,3,14,6,16,8,18,9) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 3: $C_3$ 9: $C_9$ 12: $A_4$ 36: $C_2^2 : C_9$ 576: 12T166 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: None
Degree 9: $C_9$
Low degree siblings
18T368 x 20Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 7, 8)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 7, 8)(11,12)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 9,10)(11,12)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 7, 8)( 9,10)(11,12)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 7, 8)(11,12)(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)( 9,10)(11,12)(15,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1,14, 7)( 2,13, 8)( 3,15, 9)( 4,16,10)( 5,18,11)( 6,17,12)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 7,14)( 2, 8,13)( 3, 9,15)( 4,10,16)( 5,11,18)( 6,12,17)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 7, 8)(15,16)(17,18)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(11,12)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 7, 8)(11,12)(15,16)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 9,10)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 7, 8)( 9,10)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 7, 8)( 9,10)(11,12)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 7, 8)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 7, 8)( 9,10)(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 9,10)(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5, 6)(11,12)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)(11,12)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)( 9,10)(11,12)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 5, 6)(11,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 4)( 9,10)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)( 9,10)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)( 9,10)(11,12)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(15,16)(17,18)$ |
| $ 6, 6, 3, 3 $ | $192$ | $6$ | $( 1,14, 7)( 2,13, 8)( 3,15,10, 4,16, 9)( 5,18,12, 6,17,11)$ |
| $ 6, 6, 3, 3 $ | $192$ | $6$ | $( 1, 7,14)( 2, 8,13)( 3, 9,15, 4,10,16)( 5,11,18, 6,12,17)$ |
| $ 9, 9 $ | $256$ | $9$ | $( 1,18,15,14,11, 9, 7, 5, 3)( 2,17,16,13,12,10, 8, 6, 4)$ |
| $ 9, 9 $ | $256$ | $9$ | $( 1,11, 3,14, 5,15, 7,18, 9)( 2,12, 4,13, 6,16, 8,17,10)$ |
| $ 9, 9 $ | $256$ | $9$ | $( 1, 5, 9,14,18, 3, 7,11,15)( 2, 6,10,13,17, 4, 8,12,16)$ |
| $ 9, 9 $ | $256$ | $9$ | $( 1,15,11, 7, 3,18,14, 9, 5)( 2,16,12, 8, 4,17,13,10, 6)$ |
| $ 9, 9 $ | $256$ | $9$ | $( 1, 9,18, 7,15, 5,14, 3,11)( 2,10,17, 8,16, 6,13, 4,12)$ |
| $ 9, 9 $ | $256$ | $9$ | $( 1, 3, 5, 7, 9,11,14,15,18)( 2, 4, 6, 8,10,12,13,16,17)$ |
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. |