Group action invariants
Degree $n$: | $27$ | |
Transitive number $t$: | $11$ | |
Group: | $He_3:C_2$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $3$ | |
Generators: | (1,18,4,22,25,20)(2,16,5,23,26,21)(3,17,6,24,27,19)(7,8,9)(10,15,12,14,11,13), (1,27)(2,25)(3,26)(7,19)(8,20)(9,21)(10,17)(11,18)(12,16)(13,24)(14,22)(15,23) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 9: $S_3\times C_3$, $C_3^2 : C_6$, $C_3^2 : S_3 $
Low degree siblings
9T11, 9T13, 18T20, 18T21, 18T22Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $9$ | $2$ | $( 4,26)( 5,27)( 6,25)( 7,23)( 8,24)( 9,22)(10,21)(11,19)(12,20)(13,16)(14,17) (15,18)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7,12,14)( 8,10,15)( 9,11,13)(16,22,19)(17,23,20) (18,24,21)(25,26,27)$ |
$ 6, 6, 6, 6, 3 $ | $9$ | $6$ | $( 1, 2, 3)( 4,27, 6,26, 5,25)( 7,20,14,23,12,17)( 8,21,15,24,10,18) ( 9,19,13,22,11,16)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7,14,12)( 8,15,10)( 9,13,11)(16,19,22)(17,20,23) (18,21,24)(25,27,26)$ |
$ 6, 6, 6, 6, 3 $ | $9$ | $6$ | $( 1, 3, 2)( 4,25, 5,26, 6,27)( 7,17,12,23,14,20)( 8,18,10,24,15,21) ( 9,16,11,22,13,19)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,24) (17,21,22)(18,19,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 7,16)( 2, 8,17)( 3, 9,18)( 4,10,19)( 5,11,20)( 6,12,21)(13,22,25) (14,23,26)(15,24,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 8,19)( 2, 9,20)( 3, 7,21)( 4,11,22)( 5,12,23)( 6,10,24)(13,18,27) (14,16,25)(15,17,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 9,22)( 2, 7,23)( 3, 8,24)( 4,12,16)( 5,10,17)( 6,11,18)(13,20,26) (14,21,27)(15,19,25)$ |
Group invariants
Order: | $54=2 \cdot 3^{3}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [54, 5] |
Character table: |
2 1 1 1 1 1 1 . . . . 3 3 1 2 1 2 1 3 2 2 2 1a 2a 3a 6a 3b 6b 3c 3d 3e 3f 2P 1a 1a 3b 3b 3a 3a 3c 3e 3d 3f 3P 1a 2a 1a 2a 1a 2a 1a 1a 1a 1a 5P 1a 2a 3b 6b 3a 6a 3c 3e 3d 3f X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 1 1 1 1 X.3 1 -1 A -A /A -/A 1 A /A 1 X.4 1 -1 /A -/A A -A 1 /A A 1 X.5 1 1 A A /A /A 1 A /A 1 X.6 1 1 /A /A A A 1 /A A 1 X.7 2 . 2 . 2 . 2 -1 -1 -1 X.8 2 . B . /B . 2 -/A -A -1 X.9 2 . /B . B . 2 -A -/A -1 X.10 6 . . . . . -3 . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 |