Group action invariants
| Degree $n$: | $18$ | |
| Transitive number $t$: | $25$ | |
| Group: | $C_6\times A_4$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $6$ | |
| Generators: | (1,9,15,2,10,16)(3,11,17,4,12,18)(5,7,14,6,8,13), (1,3,6,2,4,5)(7,10,11)(8,9,12)(13,16,18,14,15,17) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ x 4 $6$: $C_6$ x 4 $9$: $C_3^2$ $12$: $A_4$ $18$: $C_6 \times C_3$ $24$: $A_4\times C_2$ $36$: $C_3\times A_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$ x 4
Degree 6: $A_4\times C_2$
Degree 9: $C_3^2$
Low degree siblings
There are no siblings 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, 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, 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, 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, 3, 3 $ | $3$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7, 9,11, 8,10,12)(13,15,18)(14,16,17)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7, 9,11, 8,10,12)(13,16,18,14,15,17)$ |
| $ 6, 3, 3, 3, 3 $ | $3$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7,10,11)( 8, 9,12)(13,15,18)(14,16,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 4, 6)( 2, 3, 5)( 7,10,11)( 8, 9,12)(13,15,18)(14,16,17)$ |
| $ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 5, 4, 2, 6, 3)( 7,11,10)( 8,12, 9)(13,17,15,14,18,16)$ |
| $ 6, 3, 3, 3, 3 $ | $3$ | $6$ | $( 1, 5, 4, 2, 6, 3)( 7,11,10)( 8,12, 9)(13,18,15)(14,17,16)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1, 5, 4, 2, 6, 3)( 7,12,10, 8,11, 9)(13,17,15,14,18,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 6, 4)( 2, 5, 3)( 7,11,10)( 8,12, 9)(13,18,15)(14,17,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 7,17)( 2, 8,18)( 3, 9,13)( 4,10,14)( 5,12,15)( 6,11,16)$ |
| $ 6, 6, 6 $ | $4$ | $6$ | $( 1, 7,17, 2, 8,18)( 3, 9,13, 4,10,14)( 5,12,15, 6,11,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 9,16)( 2,10,15)( 3,11,18)( 4,12,17)( 5, 7,13)( 6, 8,14)$ |
| $ 6, 6, 6 $ | $4$ | $6$ | $( 1, 9,16, 2,10,15)( 3,11,18, 4,12,17)( 5, 7,13, 6, 8,14)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,11,13)( 2,12,14)( 3, 8,16)( 4, 7,15)( 5, 9,17)( 6,10,18)$ |
| $ 6, 6, 6 $ | $4$ | $6$ | $( 1,11,13, 2,12,14)( 3, 8,16, 4, 7,15)( 5, 9,17, 6,10,18)$ |
| $ 6, 6, 6 $ | $4$ | $6$ | $( 1,13,11, 2,14,12)( 3,16, 8, 4,15, 7)( 5,17, 9, 6,18,10)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,13,12)( 2,14,11)( 3,16, 7)( 4,15, 8)( 5,17,10)( 6,18, 9)$ |
| $ 6, 6, 6 $ | $4$ | $6$ | $( 1,15,10, 2,16, 9)( 3,17,12, 4,18,11)( 5,14, 8, 6,13, 7)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,15, 9)( 2,16,10)( 3,17,11)( 4,18,12)( 5,14, 7)( 6,13, 8)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,17, 7)( 2,18, 8)( 3,13, 9)( 4,14,10)( 5,15,12)( 6,16,11)$ |
| $ 6, 6, 6 $ | $4$ | $6$ | $( 1,17, 8, 2,18, 7)( 3,13,10, 4,14, 9)( 5,15,11, 6,16,12)$ |
Group invariants
| Order: | $72=2^{3} \cdot 3^{2}$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [72, 47] |
| Character table: not available. |