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Properties

Label 27T14
Order \(54\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_9:C_3$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $14$
Group :  $D_9:C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,27,5,21,17,22,11,7,15)(2,25,6,19,18,23,12,8,13)(3,26,4,20,16,24,10,9,14), (1,27,19,8,10,16)(2,25,20,9,11,17)(3,26,21,7,12,18)(4,6,5)(13,22,14,23,15,24)
$|\Aut(F/K)|$:  $3$

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 3: $C_3$, $S_3$

Degree 9: $S_3\times C_3$, $(C_9:C_3):C_2$

Low degree siblings

9T10, 18T18

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 $ $9$ $2$ $( 4,25)( 5,26)( 6,27)( 7,23)( 8,24)( 9,22)(10,20)(11,21)(12,19)(13,17)(14,18) (15,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 2, 3)( 4,15,23)( 5,13,24)( 6,14,22)( 7,25,16)( 8,26,17)( 9,27,18) (10,11,12)(19,20,21)$
$ 6, 6, 6, 6, 3 $ $9$ $6$ $( 1, 2, 3)( 4,16,23,25,15, 7)( 5,17,24,26,13, 8)( 6,18,22,27,14, 9) (10,21,12,20,11,19)$
$ 6, 6, 6, 6, 3 $ $9$ $6$ $( 1, 3, 2)( 4, 7,15,25,23,16)( 5, 8,13,26,24,17)( 6, 9,14,27,22,18) (10,19,11,20,12,21)$
$ 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 $ $6$ $9$ $( 1, 4, 8,11,14,18,21,24,25)( 2, 5, 9,12,15,16,19,22,26)( 3, 6, 7,10,13,17,20, 23,27)$
$ 9, 9, 9 $ $6$ $9$ $( 1, 5,17,11,15,27,21,22, 7)( 2, 6,18,12,13,25,19,23, 8)( 3, 4,16,10,14,26,20, 24, 9)$
$ 9, 9, 9 $ $6$ $9$ $( 1, 6,26,11,13, 9,21,23,16)( 2, 4,27,12,14, 7,19,24,17)( 3, 5,25,10,15, 8,20, 22,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,11,21)( 2,12,19)( 3,10,20)( 4,14,24)( 5,15,22)( 6,13,23)( 7,17,27) ( 8,18,25)( 9,16,26)$

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 6]
Character table:    
      2  1  1  1   1   1  1  .   .   .  .
      3  3  1  2   1   1  2  2   2   2  3

        1a 2a 3a  6a  6b 3b 9a  9b  9c 3c
     2P 1a 1a 3b  3b  3a 3a 9a  9c  9b 3c
     3P 1a 2a 1a  2a  2a 1a 3c  3c  3c 1a
     5P 1a 2a 3b  6b  6a 3a 9a  9c  9b 3c
     7P 1a 2a 3a  6a  6b 3b 9a  9b  9c 3c

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

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