Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $713$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,6,4,2,5,3)(7,17,9,16,8,18,10,15)(11,14)(12,13), (1,18,3,15,5,14)(2,17,4,16,6,13)(7,11)(8,12) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 6: $S_3$ 24: $S_4$ 648: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: None
Degree 9: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Low degree siblings
12T291, 18T714, 18T715, 18T721Siblings 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 $ | $9$ | $2$ | $(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 9,10)(11,12)(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 5, 6)( 9,10)(11,12)(13,14)(17,18)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $3$ | $( 1, 4, 5)( 2, 3, 6)$ |
| $ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $144$ | $6$ | $( 1, 4, 5)( 2, 3, 6)(13,14)(17,18)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $216$ | $6$ | $( 1, 4, 5)( 2, 3, 6)( 9,10)(11,12)(13,14)(17,18)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $192$ | $3$ | $( 1, 4, 5)( 2, 3, 6)(13,17,15)(14,18,16)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $576$ | $6$ | $( 1, 4, 5)( 2, 3, 6)( 9,10)(11,12)(13,17,15)(14,18,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $512$ | $3$ | $( 1, 4, 5)( 2, 3, 6)( 7,12, 9)( 8,11,10)(13,17,15)(14,18,16)$ |
| $ 4, 4, 3, 3, 1, 1, 1, 1 $ | $864$ | $12$ | $( 1, 4, 5)( 2, 3, 6)( 7,11, 8,12)(15,17,16,18)$ |
| $ 4, 3, 3, 2, 2, 2, 1, 1 $ | $1728$ | $12$ | $( 1, 4, 5)( 2, 3, 6)( 7,11, 8,12)(13,14)(15,18)(16,17)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $864$ | $6$ | $( 1, 4, 5)( 2, 3, 6)( 7,12)( 8,11)( 9,10)(13,14)(15,18)(16,17)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $108$ | $4$ | $( 7,11, 8,12)(15,17,16,18)$ |
| $ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $216$ | $4$ | $( 7,11, 8,12)(13,14)(15,18)(16,17)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $108$ | $2$ | $( 7,12)( 8,11)( 9,10)(13,14)(15,18)(16,17)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $324$ | $4$ | $( 1, 2)( 5, 6)( 7,11, 8,12)(15,17,16,18)$ |
| $ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $648$ | $4$ | $( 1, 2)( 5, 6)( 7,11, 8,12)(13,14)(15,18)(16,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $324$ | $2$ | $( 1, 2)( 5, 6)( 7,12)( 8,11)( 9,10)(13,14)(15,18)(16,17)$ |
| $ 9, 9 $ | $4608$ | $9$ | $( 1,10,14, 6,11,18, 3, 8,16)( 2, 9,13, 5,12,17, 4, 7,15)$ |
| $ 6, 6, 3, 3 $ | $3456$ | $6$ | $( 1,10,14, 2, 9,13)( 3, 8,16, 4, 7,15)( 5,12,17)( 6,11,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1152$ | $3$ | $( 1,10,13)( 2, 9,14)( 3, 8,15)( 4, 7,16)( 5,12,17)( 6,11,18)$ |
| $ 9, 9 $ | $4608$ | $9$ | $( 1,10,14, 3, 8,16, 5,12,17)( 2, 9,13, 4, 7,15, 6,11,18)$ |
| $ 6, 6, 2, 2, 1, 1 $ | $3456$ | $6$ | $( 1,11, 6, 7, 4, 9)( 2,12, 5, 8, 3,10)(15,17)(16,18)$ |
| $ 6, 6, 4, 2 $ | $3456$ | $12$ | $( 1,11, 6, 7, 4, 9)( 2,12, 5, 8, 3,10)(13,14)(15,18,16,17)$ |
| $ 4, 4, 2, 2, 2, 2, 1, 1 $ | $1296$ | $4$ | $( 1,11, 2,12)( 3,10)( 4, 9)( 5, 8, 6, 7)(15,17)(16,18)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $1296$ | $4$ | $( 1,11, 2,12)( 3,10)( 4, 9)( 5, 8, 6, 7)(13,14)(15,18,16,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $432$ | $2$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 7)( 6, 8)(15,17)(16,18)$ |
| $ 4, 2, 2, 2, 2, 2, 2, 2 $ | $432$ | $4$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 7)( 6, 8)(13,14)(15,18,16,17)$ |
| $ 8, 2, 2, 2, 1, 1, 1, 1 $ | $1296$ | $8$ | $( 1, 8, 6,11, 2, 7, 5,12)( 3,10)( 4, 9)(15,16)$ |
| $ 8, 2, 2, 2, 2, 2 $ | $432$ | $8$ | $( 1, 8, 6,11, 2, 7, 5,12)( 3,10)( 4, 9)(13,14)(15,16)(17,18)$ |
| $ 4, 4, 4, 2, 1, 1, 1, 1 $ | $1296$ | $4$ | $( 1, 8, 6,12)( 2, 7, 5,11)( 3, 9, 4,10)(15,16)$ |
| $ 4, 4, 4, 2, 2, 2 $ | $432$ | $4$ | $( 1, 8, 6,12)( 2, 7, 5,11)( 3, 9, 4,10)(13,14)(15,16)(17,18)$ |
| $ 8, 6, 2, 2 $ | $1728$ | $24$ | $( 1, 8, 6,11, 2, 7, 5,12)( 3,10)( 4, 9)(13,17,15,14,18,16)$ |
| $ 6, 4, 4, 4 $ | $1728$ | $12$ | $( 1, 8, 6,12)( 2, 7, 5,11)( 3, 9, 4,10)(13,17,15,14,18,16)$ |
| $ 8, 6, 2, 2 $ | $1728$ | $24$ | $( 1, 8, 6,11, 2, 7, 5,12)( 3,10)( 4, 9)(13,15,18,14,16,17)$ |
| $ 6, 4, 4, 4 $ | $1728$ | $12$ | $( 1, 8, 6,12)( 2, 7, 5,11)( 3, 9, 4,10)(13,15,18,14,16,17)$ |
Group invariants
| Order: | $41472=2^{9} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |