Group action invariants
| Degree $n$: | $27$ | |
| Transitive number $t$: | $176$ | |
| Group: | $C_9.(C_9\times S_3)$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,10,4,16,8,13,2,12,5,18,9,15,3,11,6,17,7,14)(19,24,20,23,21,22)(25,26), (1,21,9,23,5,25,3,19,8,24,4,26,2,20,7,22,6,27)(10,14,11,13,12,15)(16,18) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $9$: $C_9$ $18$: $S_3\times C_3$, $C_{18}$ $54$: $(C_9:C_3):C_2$, $C_9\times S_3$ $162$: 18T80 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 9: $(C_9:C_3):C_2$
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, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$ |
| $ 9, 9, 3, 3, 1, 1, 1 $ | $9$ | $9$ | $( 4, 5, 6)( 7, 9, 8)(10,13,17,11,14,18,12,15,16)(19,27,22,21,26,24,20,25,23)$ |
| $ 9, 9, 3, 3, 1, 1, 1 $ | $9$ | $9$ | $( 4, 5, 6)( 7, 9, 8)(10,14,16,11,15,17,12,13,18)(19,26,23,21,25,22,20,27,24)$ |
| $ 9, 9, 3, 3, 1, 1, 1 $ | $9$ | $9$ | $( 4, 5, 6)( 7, 9, 8)(10,15,18,11,13,16,12,14,17)(19,25,24,21,27,23,20,26,22)$ |
| $ 9, 9, 3, 3, 1, 1, 1 $ | $9$ | $9$ | $( 4, 6, 5)( 7, 8, 9)(10,16,15,12,18,14,11,17,13)(19,23,25,20,24,26,21,22,27)$ |
| $ 9, 9, 3, 3, 1, 1, 1 $ | $9$ | $9$ | $( 4, 6, 5)( 7, 8, 9)(10,17,14,12,16,13,11,18,15)(19,22,26,20,23,27,21,24,25)$ |
| $ 9, 9, 3, 3, 1, 1, 1 $ | $9$ | $9$ | $( 4, 6, 5)( 7, 8, 9)(10,18,13,12,17,15,11,16,14)(19,24,27,20,22,25,21,23,26)$ |
| $ 18, 6, 2, 1 $ | $27$ | $18$ | $( 2, 3)( 4, 7, 5, 9, 6, 8)(10,19,13,27,17,22,11,21,14,26,18,24,12,20,15,25,16, 23)$ |
| $ 18, 6, 2, 1 $ | $27$ | $18$ | $( 2, 3)( 4, 7, 5, 9, 6, 8)(10,20,14,27,16,24,11,19,15,26,17,23,12,21,13,25,18, 22)$ |
| $ 18, 6, 2, 1 $ | $27$ | $18$ | $( 2, 3)( 4, 7, 5, 9, 6, 8)(10,21,15,27,18,23,11,20,13,26,16,22,12,19,14,25,17, 24)$ |
| $ 18, 6, 2, 1 $ | $27$ | $18$ | $( 2, 3)( 4, 8, 6, 9, 5, 7)(10,22,18,25,13,21,12,23,17,26,15,19,11,24,16,27,14, 20)$ |
| $ 18, 6, 2, 1 $ | $27$ | $18$ | $( 2, 3)( 4, 8, 6, 9, 5, 7)(10,23,16,25,15,20,12,24,18,26,14,21,11,22,17,27,13, 19)$ |
| $ 18, 6, 2, 1 $ | $27$ | $18$ | $( 2, 3)( 4, 8, 6, 9, 5, 7)(10,24,17,25,14,19,12,22,16,26,13,20,11,23,18,27,15, 21)$ |
| $ 6, 6, 6, 2, 2, 2, 2, 1 $ | $27$ | $6$ | $( 2, 3)( 4, 9)( 5, 8)( 6, 7)(10,25,12,26,11,27)(13,23,15,24,14,22) (16,20,18,21,17,19)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $27$ | $2$ | $( 2, 3)( 4, 9)( 5, 8)( 6, 7)(10,26)(11,25)(12,27)(13,24)(14,23)(15,22)(16,21) (17,20)(18,19)$ |
| $ 6, 6, 6, 2, 2, 2, 2, 1 $ | $27$ | $6$ | $( 2, 3)( 4, 9)( 5, 8)( 6, 7)(10,27,11,26,12,25)(13,22,14,24,15,23) (16,19,17,21,18,20)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ |
| $ 9, 9, 9 $ | $6$ | $9$ | $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,13,16,11,14,17,12,15,18)(19,22,27,20,23,25,21, 24,26)$ |
| $ 9, 9, 9 $ | $6$ | $9$ | $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,14,18,11,15,16,12,13,17)(19,24,25,20,22,26,21, 23,27)$ |
| $ 9, 9, 9 $ | $6$ | $9$ | $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,15,17,11,13,18,12,14,16)(19,23,26,20,24,27,21, 22,25)$ |
| $ 27 $ | $18$ | $27$ | $( 1,10,19, 6,15,24, 8,17,25, 2,11,20, 4,13,22, 9,18,26, 3,12,21, 5,14,23, 7, 16,27)$ |
| $ 27 $ | $18$ | $27$ | $( 1,10,20, 6,15,22, 8,17,26, 2,11,21, 4,13,23, 9,18,27, 3,12,19, 5,14,24, 7, 16,25)$ |
| $ 27 $ | $18$ | $27$ | $( 1,10,21, 6,15,23, 8,17,27, 2,11,19, 4,13,24, 9,18,25, 3,12,20, 5,14,22, 7, 16,26)$ |
| $ 27 $ | $18$ | $27$ | $( 1,10,22, 5,15,27, 9,17,20, 2,11,23, 6,13,25, 7,18,21, 3,12,24, 4,14,26, 8, 16,19)$ |
| $ 27 $ | $18$ | $27$ | $( 1,10,23, 5,15,25, 9,17,21, 2,11,24, 6,13,26, 7,18,19, 3,12,22, 4,14,27, 8, 16,20)$ |
| $ 27 $ | $18$ | $27$ | $( 1,10,24, 5,15,26, 9,17,19, 2,11,22, 6,13,27, 7,18,20, 3,12,23, 4,14,25, 8, 16,21)$ |
| $ 27 $ | $18$ | $27$ | $( 1,10,25, 6,14,21, 8,18,24, 2,11,26, 4,15,19, 9,16,22, 3,12,27, 5,13,20, 7, 17,23)$ |
| $ 27 $ | $18$ | $27$ | $( 1,10,26, 6,14,19, 8,18,22, 2,11,27, 4,15,20, 9,16,23, 3,12,25, 5,13,21, 7, 17,24)$ |
| $ 27 $ | $18$ | $27$ | $( 1,10,27, 6,14,20, 8,18,23, 2,11,25, 4,15,21, 9,16,24, 3,12,26, 5,13,19, 7, 17,22)$ |
Group invariants
| Order: | $486=2 \cdot 3^{5}$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [486, 31] |
| Character table: not available. |