Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $176$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,2)(3,6)(4,5)(9,11)(10,12)(13,14)(15,18)(16,17), (1,9,18,2,10,17)(3,12,13,4,11,14)(5,8,16,6,7,15) | |
| $|\Aut(F/K)|$: | $2$ |
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: $S_4$, $A_4\times C_2$ x 3 36: $C_6\times S_3$ 48: $S_4\times C_2$, $C_2^2 \times A_4$ 72: 12T43, 12T45 144: 18T60, 18T61 288: 16T709 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 6: None
Degree 9: $S_3\times C_3$
Low degree siblings
18T176Siblings 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 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)(17,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,11, 9)( 8,12,10)(13,17,15)(14,18,16)$ |
| $ 4, 4, 4, 2, 1, 1, 1, 1 $ | $18$ | $4$ | $( 1, 5, 2, 6)( 7,12, 8,11)( 9,10)(13,17,14,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $18$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)( 7,11)( 8,12)(13,18)(14,17)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 4)( 5, 6)( 7, 8)(13,14)$ |
| $ 6, 6, 3, 3 $ | $24$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7,12, 9, 8,11,10)(13,18,15,14,17,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1, 5)( 2, 6)( 7,11)( 8,12)(13,17)(14,18)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $6$ | $4$ | $( 1, 6, 2, 5)( 3, 4)( 7,12, 8,11)( 9,10)(13,18,14,17)(15,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $32$ | $3$ | $( 1,18,10)( 2,17, 9)( 3,13,11)( 4,14,12)( 5,16, 7)( 6,15, 8)$ |
| $ 6, 6, 3, 3 $ | $12$ | $6$ | $( 1,14, 8, 2,13, 7)( 3,15,10)( 4,16, 9)( 5,18,12, 6,17,11)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,13, 8)( 2,14, 7)( 3,15,10)( 4,16, 9)( 5,17,12)( 6,18,11)$ |
| $ 12, 6 $ | $24$ | $12$ | $( 1,13, 7, 2,14, 8)( 3,17,10, 5,16,12, 4,18, 9, 6,15,11)$ |
| $ 6, 6, 3, 3 $ | $24$ | $6$ | $( 1,14, 7)( 2,13, 8)( 3,18, 9, 5,16,11)( 4,17,10, 6,15,12)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $32$ | $3$ | $( 1,10,18)( 2, 9,17)( 3,11,13)( 4,12,14)( 5, 7,16)( 6, 8,15)$ |
| $ 6, 6, 3, 3 $ | $12$ | $6$ | $( 1, 8,13)( 2, 7,14)( 3, 9,15, 4,10,16)( 5,11,17, 6,12,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 8,13)( 2, 7,14)( 3,10,15)( 4, 9,16)( 5,12,17)( 6,11,18)$ |
| $ 6, 6, 3, 3 $ | $24$ | $6$ | $( 1, 9,14, 4, 8,15)( 2,10,13, 3, 7,16)( 5,12,18)( 6,11,17)$ |
| $ 12, 6 $ | $24$ | $12$ | $( 1,10,13, 4, 8,16, 2, 9,14, 3, 7,15)( 5,11,18, 6,12,17)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)$ |
| $ 6, 3, 3, 3, 3 $ | $24$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7,11, 9)( 8,12,10)(13,18,15,14,17,16)$ |
| $ 4, 4, 4, 2, 2, 1, 1 $ | $18$ | $4$ | $( 1, 5, 2, 6)( 7,12, 8,11)( 9,10)(13,18,14,17)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $18$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)( 7,11)( 8,12)(13,17)(14,18)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $6$ | $4$ | $( 1, 5, 2, 6)( 7,11, 8,12)(13,17,14,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)( 7,12)( 8,11)( 9,10)(13,18)(14,17)(15,16)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 5, 6)(11,12)(17,18)$ |
| $ 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 $ | $8$ | $6$ | $( 1, 5, 4, 2, 6, 3)( 7,11,10, 8,12, 9)(13,17,16,14,18,15)$ |
| $ 6, 6, 6 $ | $32$ | $6$ | $( 1,18, 9, 2,17,10)( 3,13,12, 4,14,11)( 5,16, 8, 6,15, 7)$ |
| $ 6, 3, 3, 3, 3 $ | $12$ | $6$ | $( 1,14, 7)( 2,13, 8)( 3,15, 9, 4,16,10)( 5,18,11)( 6,17,12)$ |
| $ 6, 6, 6 $ | $4$ | $6$ | $( 1,13, 7, 2,14, 8)( 3,15, 9, 4,16,10)( 5,17,11, 6,18,12)$ |
| $ 12, 3, 3 $ | $24$ | $12$ | $( 1,13, 8)( 2,14, 7)( 3,17, 9, 6,15,12, 4,18,10, 5,16,11)$ |
| $ 6, 6, 6 $ | $24$ | $6$ | $( 1,14, 8, 2,13, 7)( 3,18,10, 6,15,11)( 4,17, 9, 5,16,12)$ |
| $ 6, 6, 6 $ | $32$ | $6$ | $( 1,10,18, 2, 9,17)( 3,11,13, 4,12,14)( 5, 7,16, 6, 8,15)$ |
| $ 6, 3, 3, 3, 3 $ | $12$ | $6$ | $( 1, 8,13, 2, 7,14)( 3, 9,15)( 4,10,16)( 5,11,17)( 6,12,18)$ |
| $ 6, 6, 6 $ | $4$ | $6$ | $( 1, 8,13, 2, 7,14)( 3,10,15, 4, 9,16)( 5,12,17, 6,11,18)$ |
| $ 6, 6, 6 $ | $24$ | $6$ | $( 1, 9,14, 3, 7,16)( 2,10,13, 4, 8,15)( 5,12,18, 6,11,17)$ |
| $ 12, 3, 3 $ | $24$ | $12$ | $( 1,10,13, 3, 7,15, 2, 9,14, 4, 8,16)( 5,11,18)( 6,12,17)$ |
Group invariants
| Order: | $576=2^{6} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [576, 8655] |
| Character table: Data not available. |