Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $370$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,12,3,18,2,11,4,17)(5,6)(7,9,15,13)(8,10,16,14), (1,14,16,9,12,5)(2,13,15,10,11,6)(3,8,18)(4,7,17), (1,15)(2,16)(5,10,6,9)(7,18,8,17)(11,12) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_2^2$ x 7 8: $D_{4}$ x 2, $C_2^3$ 16: $D_4\times C_2$ 72: $C_3^2:D_4$ 144: 12T77 1152: $S_4\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: None
Degree 6: None
Degree 9: $S_3^2:C_2$
Low degree siblings
12T235 x 2, 12T236 x 2, 16T1493 x 2, 16T1494 x 2, 18T370 x 3Siblings 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 $ | $9$ | $2$ | $( 1, 2)( 3, 4)(11,12)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 5, 4)( 2, 6, 3)( 7,11, 9)( 8,12,10)(13,17,16)(14,18,15)$ |
| $ 6, 6, 3, 3 $ | $48$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7,12,10, 8,11, 9)(13,18,15,14,17,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1,11,15)( 2,12,16)( 3, 8,17)( 4, 7,18)( 5, 9,14)( 6,10,13)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $36$ | $4$ | $( 3, 6, 4, 5)( 7,14, 8,13)( 9,18)(10,17)(11,16)(12,15)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $72$ | $4$ | $( 1, 2)( 3, 6)( 4, 5)( 7,14, 8,13)( 9,17,10,18)(11,16,12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $36$ | $2$ | $( 3, 6)( 4, 5)( 7,14)( 8,13)( 9,18)(10,17)(11,15)(12,16)$ |
| $ 4, 4, 4, 4, 1, 1 $ | $144$ | $4$ | $( 3, 7, 5,13)( 4, 8, 6,14)( 9,12,18,16)(10,11,17,15)$ |
| $ 8, 4, 4, 2 $ | $144$ | $8$ | $( 1, 2)( 3, 7, 5,13, 4, 8, 6,14)( 9,11,18,16)(10,12,17,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $12$ | $2$ | $( 7,14)( 8,13)( 9,15)(10,16)(11,18)(12,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $36$ | $2$ | $( 1, 2)( 3, 4)( 7,14)( 8,13)( 9,15)(10,16)(11,17)(12,18)$ |
| $ 4, 4, 4, 2, 1, 1, 1, 1 $ | $36$ | $4$ | $( 3, 4)( 7,14, 8,13)( 9,16,10,15)(11,18,12,17)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $12$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7,14, 8,13)( 9,15,10,16)(11,18,12,17)$ |
| $ 6, 6, 3, 3 $ | $96$ | $6$ | $( 1, 5, 4)( 2, 6, 3)( 7,18, 9,14,11,15)( 8,17,10,13,12,16)$ |
| $ 12, 6 $ | $96$ | $12$ | $( 1, 5, 3, 2, 6, 4)( 7,18, 9,14,12,15, 8,17,10,13,11,16)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $72$ | $4$ | $( 3, 7, 4, 8)( 5,13, 6,14)( 9,10)(11,16)(12,15)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $24$ | $2$ | $( 3, 8)( 4, 7)( 5,13)( 6,14)(11,16)(12,15)$ |
| $ 6, 6, 3, 3 $ | $192$ | $6$ | $( 1, 5,17,15,10, 7)( 2, 6,18,16, 9, 8)( 3,11,13)( 4,12,14)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)(11,12)(13,14)(15,16)$ |
| $ 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, 3, 3, 3, 3 $ | $48$ | $6$ | $( 1, 5, 4)( 2, 6, 3)( 7,11, 9)( 8,12,10)(13,18,16,14,17,15)$ |
| $ 6, 6, 6 $ | $16$ | $6$ | $( 1, 5, 4, 2, 6, 3)( 7,11, 9, 8,12,10)(13,17,16,14,18,15)$ |
| $ 6, 6, 6 $ | $64$ | $6$ | $( 1,11,15, 2,12,16)( 3, 8,17, 4, 7,18)( 5, 9,14, 6,10,13)$ |
| $ 4, 4, 4, 2, 2, 1, 1 $ | $72$ | $4$ | $( 3, 6, 4, 5)( 7,14)( 8,13)( 9,18,10,17)(11,16,12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $36$ | $2$ | $( 1, 2)( 3, 6)( 4, 5)( 7,14)( 8,13)( 9,17)(10,18)(11,16)(12,15)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $36$ | $4$ | $( 1, 2)( 3, 6, 4, 5)( 7,14, 8,13)( 9,17)(10,18)(11,15)(12,16)$ |
| $ 8, 4, 4, 1, 1 $ | $144$ | $8$ | $( 3, 7, 5,13, 4, 8, 6,14)( 9,12,18,15)(10,11,17,16)$ |
| $ 4, 4, 4, 4, 2 $ | $144$ | $4$ | $( 1, 2)( 3, 7, 5,13)( 4, 8, 6,14)( 9,11,18,15)(10,12,17,16)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $12$ | $4$ | $( 7,14, 8,13)( 9,15,10,16)(11,18,12,17)$ |
| $ 4, 4, 4, 2, 2, 1, 1 $ | $36$ | $4$ | $( 1, 2)( 3, 4)( 7,14, 8,13)( 9,15,10,16)(11,17,12,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $36$ | $2$ | $( 3, 4)( 7,14)( 8,13)( 9,16)(10,15)(11,18)(12,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,14)( 8,13)( 9,15)(10,16)(11,18)(12,17)$ |
| $ 12, 3, 3 $ | $96$ | $12$ | $( 1, 5, 4)( 2, 6, 3)( 7,18,10,13,11,15, 8,17, 9,14,12,16)$ |
| $ 6, 6, 6 $ | $96$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,18,10,13,12,15)( 8,17, 9,14,11,16)$ |
| $ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ | $72$ | $4$ | $( 3, 7, 4, 8)( 5,13)( 6,14)( 9,10)(11,16,12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $24$ | $2$ | $( 1, 2)( 3, 7)( 4, 8)( 5,13)( 6,14)( 9,10)(11,16)(12,15)(17,18)$ |
| $ 6, 6, 6 $ | $192$ | $6$ | $( 1, 5,17,16,10, 7)( 2, 6,18,15, 9, 8)( 3,11,13, 4,12,14)$ |
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. |