Properties

Label 27T6
Order \(54\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2:S_3$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $6$
Group :  $C_3^2:S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,3)(4,6)(7,16)(8,18)(9,17)(10,21)(11,20)(12,19)(13,23)(14,22)(15,24)(25,26), (1,10,21)(2,11,19)(3,12,20)(4,13,24)(5,14,22)(6,15,23)(7,18,25)(8,16,26)(9,17,27), (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)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 4

Degree 9: $C_3^2:C_2$, $(C_3^2:C_3):C_2$ x 4

Low degree siblings

9T12 x 4, 18T24 x 4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 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 $ $9$ $2$ $( 2, 3)( 4, 6)( 7,16)( 8,18)( 9,17)(10,21)(11,20)(12,19)(13,23)(14,22)(15,24) (25,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7,12,14)( 8,10,15)( 9,11,13)(16,22,19)(17,23,20) (18,24,21)(25,26,27)$
$ 6, 6, 6, 6, 3 $ $9$ $6$ $( 1, 4,27, 3, 5,26)( 2, 6,25)( 7,23,15,19,11,18)( 8,22,13,21,12,17) ( 9,24,14,20,10,16)$
$ 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 $ $6$ $3$ $( 1, 7,16)( 2, 8,17)( 3, 9,18)( 4,10,19)( 5,11,20)( 6,12,21)(13,22,25) (14,23,26)(15,24,27)$
$ 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 $ $6$ $3$ $( 1, 9,22)( 2, 7,23)( 3, 8,24)( 4,12,16)( 5,10,17)( 6,11,18)(13,20,26) (14,21,27)(15,19,25)$
$ 6, 6, 6, 6, 3 $ $9$ $6$ $( 1, 9, 5,10,27,14)( 2, 8, 6,12,25,13)( 3, 7, 4,11,26,15)(16,23,20,18,24,19) (17,22,21)$
$ 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:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 8]
Character table:   
      2  1  1  .   1  1  .  .  .   1  1
      3  3  1  2   1  3  2  2  2   1  3

        1a 2a 3a  6a 3b 3c 3d 3e  6b 3f
     2P 1a 1a 3a  3f 3f 3c 3d 3e  3b 3b
     3P 1a 2a 1a  2a 1a 1a 1a 1a  2a 1a
     5P 1a 2a 3a  6b 3f 3c 3d 3e  6a 3b

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      2  .  2   .  2 -1 -1 -1   .  2
X.4      2  . -1   .  2  2 -1 -1   .  2
X.5      2  . -1   .  2 -1 -1  2   .  2
X.6      2  . -1   .  2 -1  2 -1   .  2
X.7      3 -1  .   A  B  .  .  .  /A /B
X.8      3 -1  .  /A /B  .  .  .   A  B
X.9      3  1  . -/A /B  .  .  .  -A  B
X.10     3  1  .  -A  B  .  .  . -/A /B

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3
B = 3*E(3)
  = (-3+3*Sqrt(-3))/2 = 3b3