Group action invariants
| Degree $n$: | $27$ | |
| Transitive number $t$: | $40$ | |
| Group: | $C_3^2:S_3:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $9$ | |
| Generators: | (1,23,15)(2,24,13)(3,22,14)(4,26,17)(5,27,18)(6,25,16)(7,21,12)(8,19,10)(9,20,11), (1,25)(2,26)(3,27)(4,21)(5,19)(6,20)(7,22)(8,23)(9,24) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $54$: $C_3^2 : C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 9: $S_3\times C_3$
Low degree siblings
27T49Siblings 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, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(10,25)(11,26)(12,27)(13,21)(14,19)(15,20)(16,24)(17,22)(18,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ |
| $ 6, 6, 6, 3, 3, 3 $ | $9$ | $6$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,25,12,27,11,26)(13,21,15,20,14,19) (16,24,18,23,17,22)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ |
| $ 6, 6, 6, 3, 3, 3 $ | $9$ | $6$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,25,11,26,12,27)(13,21,14,19,15,20) (16,24,17,22,18,23)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,13,18,11,14,16,12,15,17)(19,23,25,20,24,26,21, 22,27)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,14,17,11,15,18,12,13,16)(19,22,26,20,23,27,21, 24,25)$ |
| $ 18, 9 $ | $9$ | $18$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,19,17,26,15,23,12,21,16,25,14,22,11,20,18,27, 13,24)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,15,16,11,13,17,12,14,18)(19,23,25,20,24,26,21, 22,27)$ |
| $ 18, 9 $ | $9$ | $18$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,19,16,25,13,24,12,21,18,27,15,23,11,20,17,26, 14,22)$ |
| $ 18, 9 $ | $9$ | $18$ | $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,19,18,27,14,22,12,21,17,26,13,24,11,20,16,25, 15,23)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7, 6, 3, 9, 5, 2, 8, 4)(10,16,13,12,18,15,11,17,14)(19,25,24,21,27,23,20, 26,22)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7, 6, 3, 9, 5, 2, 8, 4)(10,17,15,12,16,14,11,18,13)(19,27,22,21,26,24,20, 25,23)$ |
| $ 18, 9 $ | $9$ | $18$ | $( 1, 7, 6, 3, 9, 5, 2, 8, 4)(10,22,13,25,18,21,11,23,14,26,16,19,12,24,15,27, 17,20)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 8, 5, 3, 7, 4, 2, 9, 6)(10,16,13,12,18,15,11,17,14)(19,27,22,21,26,24,20, 25,23)$ |
| $ 18, 9 $ | $9$ | $18$ | $( 1, 8, 5, 3, 7, 4, 2, 9, 6)(10,22,15,27,16,19,11,23,13,25,17,20,12,24,14,26, 18,21)$ |
| $ 18, 9 $ | $9$ | $18$ | $( 1, 9, 4, 3, 8, 6, 2, 7, 5)(10,22,14,26,17,20,11,23,15,27,18,21,12,24,13,25, 16,19)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1,10,25)( 2,11,26)( 3,12,27)( 4,13,21)( 5,14,19)( 6,15,20)( 7,17,22) ( 8,18,23)( 9,16,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1,13,24)( 2,14,22)( 3,15,23)( 4,18,27)( 5,16,25)( 6,17,26)( 7,10,19) ( 8,11,20)( 9,12,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1,16,19)( 2,17,20)( 3,18,21)( 4,12,24)( 5,10,22)( 6,11,23)( 7,14,26) ( 8,15,27)( 9,13,25)$ |
Group invariants
| Order: | $162=2 \cdot 3^{4}$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [162, 14] |
| Character table: not available. |