Group action invariants
Degree $n$: | $27$ | |
Transitive number $t$: | $5$ | |
Group: | $C_9:C_3$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $2$ | |
$|\Aut(F/K)|$: | $27$ | |
Generators: | (1,4,8,11,14,18,21,24,25)(2,5,9,12,15,16,19,22,26)(3,6,7,10,13,17,20,23,27), (1,12,20)(2,10,21)(3,11,19)(4,22,13)(5,23,14)(6,24,15)(7,8,9)(16,17,18)(25,26,27) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$ x 4
Low degree siblings
9T6Siblings 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, 3, 3, 3 $ | $3$ | $3$ | $( 1, 2, 3)( 4,15,23)( 5,13,24)( 6,14,22)( 7,25,16)( 8,26,17)( 9,27,18) (10,11,12)(19,20,21)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 3, 2)( 4,23,15)( 5,24,13)( 6,22,14)( 7,16,25)( 8,17,26)( 9,18,27) (10,12,11)(19,21,20)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 4, 8,11,14,18,21,24,25)( 2, 5, 9,12,15,16,19,22,26)( 3, 6, 7,10,13,17,20, 23,27)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 5,17,11,15,27,21,22, 7)( 2, 6,18,12,13,25,19,23, 8)( 3, 4,16,10,14,26,20, 24, 9)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 6,26,11,13, 9,21,23,16)( 2, 4,27,12,14, 7,19,24,17)( 3, 5,25,10,15, 8,20, 22,18)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7,22,21,27,15,11,17, 5)( 2, 8,23,19,25,13,12,18, 6)( 3, 9,24,20,26,14,10, 16, 4)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 8,14,21,25, 4,11,18,24)( 2, 9,15,19,26, 5,12,16,22)( 3, 7,13,20,27, 6,10, 17,23)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 9, 6,21,26,23,11,16,13)( 2, 7, 4,19,27,24,12,17,14)( 3, 8, 5,20,25,22,10, 18,15)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,11,21)( 2,12,19)( 3,10,20)( 4,14,24)( 5,15,22)( 6,13,23)( 7,17,27) ( 8,18,25)( 9,16,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,21,11)( 2,19,12)( 3,20,10)( 4,24,14)( 5,22,15)( 6,23,13)( 7,27,17) ( 8,25,18)( 9,26,16)$ |
Group invariants
Order: | $27=3^{3}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [27, 4] |
Character table: |
3 3 2 2 2 2 2 2 2 2 3 3 1a 3a 3b 9a 9b 9c 9d 9e 9f 3c 3d 2P 1a 3b 3a 9e 9d 9f 9b 9a 9c 3d 3c 3P 1a 1a 1a 3c 3c 3c 3d 3d 3d 1a 1a 5P 1a 3b 3a 9e 9d 9f 9b 9a 9c 3d 3c 7P 1a 3a 3b 9a 9b 9c 9d 9e 9f 3c 3d X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 A A A /A /A /A 1 1 X.3 1 1 1 /A /A /A A A A 1 1 X.4 1 A /A 1 A /A /A 1 A 1 1 X.5 1 /A A 1 /A A A 1 /A 1 1 X.6 1 A /A A /A 1 A /A 1 1 1 X.7 1 /A A /A A 1 /A A 1 1 1 X.8 1 A /A /A 1 A 1 A /A 1 1 X.9 1 /A A A 1 /A 1 /A A 1 1 X.10 3 . . . . . . . . B /B X.11 3 . . . . . . . . /B B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 |