Group action invariants
Degree $n$: | $27$ | |
Transitive number $t$: | $3$ | |
Group: | $He_3$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $2$ | |
$|\Aut(F/K)|$: | $27$ | |
Generators: | (1,21,10)(2,19,11)(3,20,12)(4,24,13)(5,22,14)(6,23,15)(7,25,18)(8,26,16)(9,27,17), (1,25,4)(2,26,5)(3,27,6)(7,8,9)(10,11,12)(13,14,15)(16,21,23)(17,19,24)(18,20,22) |
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
Degree 9: $C_3^2$, $C_3^2:C_3$ x 4
Low degree siblings
9T7 x 4Siblings 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 $ | $3$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7,12,14)( 8,10,15)( 9,11,13)(16,22,19)(17,23,20) (18,24,21)(25,26,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7,14,12)( 8,15,10)( 9,13,11)(16,19,22)(17,20,23) (18,21,24)(25,27,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,24) (17,21,22)(18,19,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 7,16)( 2, 8,17)( 3, 9,18)( 4,10,19)( 5,11,20)( 6,12,21)(13,22,25) (14,23,26)(15,24,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 8,19)( 2, 9,20)( 3, 7,21)( 4,11,22)( 5,12,23)( 6,10,24)(13,18,27) (14,16,25)(15,17,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 9,22)( 2, 7,23)( 3, 8,24)( 4,12,16)( 5,10,17)( 6,11,18)(13,20,26) (14,21,27)(15,19,25)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1,16, 7)( 2,17, 8)( 3,18, 9)( 4,19,10)( 5,20,11)( 6,21,12)(13,25,22) (14,26,23)(15,27,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1,17,14)( 2,18,15)( 3,16,13)( 4,20, 8)( 5,21, 9)( 6,19, 7)(10,27,22) (11,25,23)(12,26,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1,18,12)( 2,16,10)( 3,17,11)( 4,21,15)( 5,19,13)( 6,20,14)( 7,26,22) ( 8,27,23)( 9,25,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,27, 5)( 2,25, 6)( 3,26, 4)( 7,15,11)( 8,13,12)( 9,14,10)(16,24,20) (17,22,21)(18,23,19)$ |
Group invariants
Order: | $27=3^{3}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [27, 3] |
Character table: |
3 3 2 2 3 2 2 2 2 2 2 3 1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j 2P 1a 3b 3a 3j 3g 3i 3h 3d 3f 3e 3c 3P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 A A A /A /A /A 1 X.3 1 1 1 1 /A /A /A A A A 1 X.4 1 A /A 1 1 A /A 1 A /A 1 X.5 1 /A A 1 1 /A A 1 /A A 1 X.6 1 A /A 1 A /A 1 /A 1 A 1 X.7 1 /A A 1 /A A 1 A 1 /A 1 X.8 1 A /A 1 /A 1 A A /A 1 1 X.9 1 /A A 1 A 1 /A /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 |