Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $263$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (5,6)(11,12)(13,14)(15,16)(17,18), (1,3,5,2,4,6)(7,9,12,8,10,11)(13,16,17,14,15,18), (1,12,14)(2,11,13)(3,7,15,4,8,16)(5,9,17,6,10,18) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ x 4 6: $C_6$ x 4 9: $C_3^2$ 12: $A_4$ x 3 18: $C_6 \times C_3$ 24: $A_4\times C_2$ x 3 36: $C_3\times A_4$ x 3 72: 18T25 x 3 144: 12T85 x 3 288: 18T109 x 3 576: 12T164 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$ x 4
Degree 6: None
Degree 9: $C_3^2$
Low degree siblings
18T263 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5, 6)(11,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 1, 2)( 9,10)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 1, 2)( 5, 6)( 9,10)(11,12)(13,14)(17,18)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 4)( 7, 8)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)$ |
| $ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5, 6)(11,12)(17,18)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 1, 2)( 7, 8)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 4)( 9,10)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 7, 8)( 9,10)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$ |
| $ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)(11,12)(13,14)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 5, 4)( 2, 6, 3)( 7,12,10)( 8,11, 9)(13,17,15)(14,18,16)$ |
| $ 6, 6, 6 $ | $64$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7,11, 9, 8,12,10)(13,18,15,14,17,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 4, 5)( 2, 3, 6)( 7,10,12)( 8, 9,11)(13,15,17)(14,16,18)$ |
| $ 6, 6, 6 $ | $64$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,10,11, 8, 9,12)(13,16,17,14,15,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $16$ | $3$ | $( 1,14,12)( 2,13,11)( 3,15, 8)( 4,16, 7)( 5,17,10)( 6,18, 9)$ |
| $ 6, 3, 3, 3, 3 $ | $48$ | $6$ | $( 1,13,12)( 2,14,11)( 3,16, 7, 4,15, 8)( 5,18, 9)( 6,17,10)$ |
| $ 6, 6, 3, 3 $ | $48$ | $6$ | $( 1,14,11)( 2,13,12)( 3,15, 7, 4,16, 8)( 5,17,10, 6,18, 9)$ |
| $ 6, 6, 6 $ | $16$ | $6$ | $( 1,13,12, 2,14,11)( 3,16, 8, 4,15, 7)( 5,18,10, 6,17, 9)$ |
| $ 6, 6, 3, 3 $ | $48$ | $6$ | $( 1,18, 8, 2,17, 7)( 3,13, 9)( 4,14,10)( 5,15,11, 6,16,12)$ |
| $ 6, 3, 3, 3, 3 $ | $48$ | $6$ | $( 1,17, 7)( 2,18, 8)( 3,14,10, 4,13, 9)( 5,16,11)( 6,15,12)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $16$ | $3$ | $( 1,17, 7)( 2,18, 8)( 3,13, 9)( 4,14,10)( 5,16,12)( 6,15,11)$ |
| $ 6, 6, 6 $ | $16$ | $6$ | $( 1,18, 8, 2,17, 7)( 3,14,10, 4,13, 9)( 5,15,12, 6,16,11)$ |
| $ 6, 6, 3, 3 $ | $48$ | $6$ | $( 1,16,10)( 2,15, 9)( 3,17,12, 4,18,11)( 5,13, 8, 6,14, 7)$ |
| $ 6, 6, 6 $ | $16$ | $6$ | $( 1,15, 9, 2,16,10)( 3,18,12, 4,17,11)( 5,14, 7, 6,13, 8)$ |
| $ 6, 3, 3, 3, 3 $ | $48$ | $6$ | $( 1,15, 9)( 2,16,10)( 3,18,11)( 4,17,12)( 5,14, 8, 6,13, 7)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $16$ | $3$ | $( 1,16, 9)( 2,15,10)( 3,17,12)( 4,18,11)( 5,13, 7)( 6,14, 8)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $16$ | $3$ | $( 1,12,14)( 2,11,13)( 3, 8,15)( 4, 7,16)( 5,10,17)( 6, 9,18)$ |
| $ 6, 3, 3, 3, 3 $ | $48$ | $6$ | $( 1,11,14)( 2,12,13)( 3, 8,16, 4, 7,15)( 5,10,18)( 6, 9,17)$ |
| $ 6, 6, 3, 3 $ | $48$ | $6$ | $( 1,11,13)( 2,12,14)( 3, 7,16, 4, 8,15)( 5,10,17, 6, 9,18)$ |
| $ 6, 6, 6 $ | $16$ | $6$ | $( 1,11,14, 2,12,13)( 3, 7,15, 4, 8,16)( 5, 9,17, 6,10,18)$ |
| $ 6, 6, 3, 3 $ | $48$ | $6$ | $( 1,10,15, 2, 9,16)( 3,11,17, 4,12,18)( 5, 7,14)( 6, 8,13)$ |
| $ 6, 3, 3, 3, 3 $ | $48$ | $6$ | $( 1,10,16)( 2, 9,15)( 3,12,17, 4,11,18)( 5, 7,13)( 6, 8,14)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $16$ | $3$ | $( 1,10,15)( 2, 9,16)( 3,12,18)( 4,11,17)( 5, 8,13)( 6, 7,14)$ |
| $ 6, 6, 6 $ | $16$ | $6$ | $( 1, 9,15, 2,10,16)( 3,11,18, 4,12,17)( 5, 7,13, 6, 8,14)$ |
| $ 6, 6, 3, 3 $ | $48$ | $6$ | $( 1, 7,18, 2, 8,17)( 3, 9,14, 4,10,13)( 5,12,16)( 6,11,15)$ |
| $ 6, 3, 3, 3, 3 $ | $48$ | $6$ | $( 1, 7,17)( 2, 8,18)( 3, 9,13)( 4,10,14)( 5,11,16, 6,12,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 7,17)( 2, 8,18)( 3, 9,13)( 4,10,14)( 5,12,16)( 6,11,15)$ |
| $ 6, 6, 6 $ | $16$ | $6$ | $( 1, 7,18, 2, 8,17)( 3, 9,14, 4,10,13)( 5,11,16, 6,12,15)$ |
Group invariants
| Order: | $1152=2^{7} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |