Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $366$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,7,18)(2,8,17)(3,9,13)(4,10,14)(5,12,15)(6,11,16), (1,5,3,2,6,4)(7,12,10,8,11,9)(13,17,15,14,18,16), (1,10,6,7,4,11)(2,9,5,8,3,12)(13,15,17)(14,16,18) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $S_3$, $C_6$ x 3 12: $A_4$, $D_{6}$, $C_6\times C_2$ 18: $S_3\times C_3$ 24: $A_4\times C_2$ x 3 36: $C_6\times S_3$ 48: $C_2^2 \times A_4$ 72: 12T43 144: 18T60 288: $A_4\wr C_2$ 576: 12T158 1152: 12T206 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 6: None
Degree 9: $S_3\times C_3$
Low degree siblings
18T366 x 5Siblings 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, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $32$ | $3$ | $( 1, 7,18)( 2, 8,17)( 3, 9,13)( 4,10,14)( 5,12,15)( 6,11,16)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)(11,12)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 1, 2)( 5, 6)( 9,10)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,14)(17,18)$ |
| $ 6, 6, 3, 3 $ | $96$ | $6$ | $( 1, 7,18, 2, 8,17)( 3, 9,13, 4,10,14)( 5,12,15)( 6,11,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,10,11)( 8, 9,12)(13,15,18)(14,16,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $32$ | $3$ | $( 1, 9,15)( 2,10,16)( 3,11,18)( 4,12,17)( 5, 8,13)( 6, 7,14)$ |
| $ 6, 6, 3, 3 $ | $96$ | $6$ | $( 1,10,15)( 2, 9,16)( 3,12,17, 4,11,18)( 5, 8,14, 6, 7,13)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,11,10)( 8,12, 9)(13,18,15)(14,17,16)$ |
| $ 6, 6, 3, 3 $ | $96$ | $6$ | $( 1,11,14, 2,12,13)( 3, 7,16, 4, 8,15)( 5,10,18)( 6, 9,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $32$ | $3$ | $( 1,12,14)( 2,11,13)( 3, 8,16)( 4, 7,15)( 5,10,18)( 6, 9,17)$ |
| $ 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, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(11,12)(15,16)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 9,10)(13,14)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 1, 2)( 7, 8)(17,18)$ |
| $ 6, 3, 3, 3, 3 $ | $96$ | $6$ | $( 1, 7,18, 2, 8,17)( 3, 9,13)( 4,10,14)( 5,11,15)( 6,12,16)$ |
| $ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1, 2)( 9,10)(15,16)$ |
| $ 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 $ | $32$ | $6$ | $( 1, 7,18, 2, 8,17)( 3, 9,13, 4,10,14)( 5,11,15, 6,12,16)$ |
| $ 6, 6, 6 $ | $64$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7,10,11, 8, 9,12)(13,15,18,14,16,17)$ |
| $ 6, 3, 3, 3, 3 $ | $96$ | $6$ | $( 1, 9,15)( 2,10,16)( 3,11,17)( 4,12,18)( 5, 7,14, 6, 8,13)$ |
| $ 6, 6, 6 $ | $32$ | $6$ | $( 1,10,16, 2, 9,15)( 3,11,18, 4,12,17)( 5, 8,14, 6, 7,13)$ |
| $ 6, 6, 6 $ | $64$ | $6$ | $( 1, 6, 4, 2, 5, 3)( 7,11, 9, 8,12,10)(13,18,16,14,17,15)$ |
| $ 6, 3, 3, 3, 3 $ | $96$ | $6$ | $( 1,11,13)( 2,12,14)( 3, 7,16, 4, 8,15)( 5, 9,17)( 6,10,18)$ |
| $ 6, 6, 6 $ | $32$ | $6$ | $( 1,12,13, 2,11,14)( 3, 7,16, 4, 8,15)( 5,10,17, 6, 9,18)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $4$ | $( 7,17)( 8,18)( 9,14,10,13)(11,16,12,15)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $36$ | $4$ | $( 1, 2)( 3, 4)( 7,17, 8,18)( 9,13)(10,14)(11,15,12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $36$ | $2$ | $( 3, 4)( 5, 6)( 7,18)( 8,17)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $36$ | $4$ | $( 1, 2)( 5, 6)( 7,18, 8,17)( 9,13,10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $12$ | $2$ | $( 7,17)( 8,18)( 9,13)(10,14)(11,15)(12,16)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $36$ | $4$ | $( 1, 2)( 3, 4)( 7,17, 8,18)( 9,14,10,13)(11,16)(12,15)$ |
| $ 6, 6, 3, 3 $ | $192$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7,14,11,17,10,15)( 8,13,12,18, 9,16)$ |
| $ 6, 6, 3, 3 $ | $192$ | $6$ | $( 1, 6, 3)( 2, 5, 4)( 7,16, 9,17,11,14)( 8,15,10,18,12,13)$ |
| $ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ | $36$ | $4$ | $( 5, 6)( 7,18, 8,17)( 9,14,10,13)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,18)( 8,17)( 9,13)(10,14)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $36$ | $2$ | $( 5, 6)( 7,17)( 8,18)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 4, 4, 2, 2, 2, 2, 2 $ | $36$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7,17, 8,18)( 9,14,10,13)(11,15)(12,16)$ |
| $ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ | $36$ | $4$ | $( 3, 4)( 7,18)( 8,17)( 9,13,10,14)(11,15,12,16)$ |
| $ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ | $36$ | $4$ | $( 1, 2)( 7,18, 8,17)( 9,14)(10,13)(11,16,12,15)$ |
| $ 6, 6, 6 $ | $192$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7,13,12,18, 9,16)( 8,14,11,17,10,15)$ |
| $ 6, 6, 6 $ | $192$ | $6$ | $( 1, 6, 4, 2, 5, 3)( 7,15, 9,17,11,14)( 8,16,10,18,12,13)$ |
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. |