Group action invariants
Degree $n$: | $27$ | |
Transitive number $t$: | $23$ | |
Group: | $He_3:C_3$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $3$ | |
$|\Aut(F/K)|$: | $3$ | |
Generators: | (4,5,6)(7,15,10)(8,13,11)(9,14,12)(16,20,22)(17,21,23)(18,19,24)(25,27,26), (1,11,20,2,12,21,3,10,19)(4,15,23,5,13,24,6,14,22)(7,17,26,8,18,27,9,16,25) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ $27$: $C_3^2:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 9: $C_3^2:C_3$
Low degree siblings
27T23 x 2, 27T24Siblings 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, 1, 1, 1 $ | $9$ | $3$ | $( 4, 5, 6)( 7,15,10)( 8,13,11)( 9,14,12)(16,20,22)(17,21,23)(18,19,24) (25,27,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $9$ | $3$ | $( 4, 6, 5)( 7,10,15)( 8,11,13)( 9,12,14)(16,22,20)(17,23,21)(18,24,19) (25,26,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,11,15)( 8,12,13)( 9,10,14)(16,19,22) (17,20,23)(18,21,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $9$ | $3$ | $( 1, 7,16)( 2, 8,17)( 3, 9,18)( 4,10,20)( 5,11,21)( 6,12,19)(13,24,25) (14,22,26)(15,23,27)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7,23, 2, 8,24, 3, 9,22)( 4,11,17, 5,12,18, 6,10,16)(13,21,27,14,19,25,15, 20,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $9$ | $3$ | $( 1, 7,20)( 2, 8,21)( 3, 9,19)( 4,12,22)( 5,10,23)( 6,11,24)(13,17,27) (14,18,25)(15,16,26)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 8,22, 2, 9,23, 3, 7,24)( 4,12,16, 5,10,17, 6,11,18)(13,19,26,14,20,27,15, 21,25)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 9,24, 2, 7,22, 3, 8,23)( 4,10,18, 5,11,16, 6,12,17)(13,20,25,14,21,26,15, 19,27)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1,16,14, 3,18,13, 2,17,15)( 4,19, 9, 6,21, 8, 5,20, 7)(10,27,24,12,26,23,11, 25,22)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $9$ | $3$ | $( 1,16, 7)( 2,17, 8)( 3,18, 9)( 4,20,10)( 5,21,11)( 6,19,12)(13,25,24) (14,26,22)(15,27,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $9$ | $3$ | $( 1,16,10)( 2,17,11)( 3,18,12)( 4,21,15)( 5,19,13)( 6,20,14)( 7,27,22) ( 8,25,23)( 9,26,24)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1,17,13, 3,16,15, 2,18,14)( 4,20, 8, 6,19, 7, 5,21, 9)(10,25,23,12,27,22,11, 26,24)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1,18,15, 3,17,14, 2,16,13)( 4,21, 7, 6,20, 9, 5,19, 8)(10,26,22,12,25,24,11, 27,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1,25, 4)( 2,26, 5)( 3,27, 6)( 7,15,11)( 8,13,12)( 9,14,10)(16,22,19) (17,23,20)(18,24,21)$ |
Group invariants
Order: | $81=3^{4}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [81, 9] |
Character table: |
3 4 2 2 4 4 3 2 3 2 3 3 3 2 2 3 3 3 1a 3a 3b 3c 3d 3e 3f 9a 3g 9b 9c 9d 3h 3i 9e 9f 3j 2P 1a 3b 3a 3d 3c 3j 3h 9e 3i 9d 9f 9c 3f 3g 9b 9a 3e 3P 1a 1a 1a 1a 1a 1a 1a 3c 1a 3c 3c 3d 1a 1a 3d 3d 1a 5P 1a 3b 3a 3d 3c 3j 3h 9f 3i 9e 9d 9b 3f 3g 9a 9c 3e 7P 1a 3a 3b 3c 3d 3e 3f 9c 3g 9a 9b 9e 3h 3i 9f 9d 3j X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 A A A A A /A /A /A /A /A 1 X.3 1 1 1 1 1 1 /A /A /A /A /A A A A A A 1 X.4 1 A /A 1 1 1 1 A /A A A /A 1 A /A /A 1 X.5 1 /A A 1 1 1 1 /A A /A /A A 1 /A A A 1 X.6 1 A /A 1 1 1 A /A 1 /A /A A /A 1 A A 1 X.7 1 /A A 1 1 1 /A A 1 A A /A A 1 /A /A 1 X.8 1 A /A 1 1 1 /A 1 A 1 1 1 A /A 1 1 1 X.9 1 /A A 1 1 1 A 1 /A 1 1 1 /A A 1 1 1 X.10 3 . . 3 3 B . . . . . . . . . . /B X.11 3 . . 3 3 /B . . . . . . . . . . B X.12 3 . . B /B . . C . D E /C . . /E /D . X.13 3 . . B /B . . D . E C /D . . /C /E . X.14 3 . . B /B . . E . C D /E . . /D /C . X.15 3 . . /B B . . /C . /D /E C . . E D . X.16 3 . . /B B . . /E . /C /D E . . D C . X.17 3 . . /B B . . /D . /E /C D . . C E . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 C = -E(9)^2-2*E(9)^5 D = -E(9)^2+E(9)^5 E = 2*E(9)^2+E(9)^5 |