Properties

Label 9T6
Order \(27\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_9:C_3$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $6$
Group :  $C_9:C_3$
CHM label :  $1/3[3^{3}]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,4,7)(2,8,5), (1,2,3,4,5,6,7,8,9)
$|\Aut(F/K)|$:  $3$

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: $C_3$

Low degree siblings

27T5

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 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1 $ $3$ $3$ $(2,5,8)(3,9,6)$
$ 3, 3, 1, 1, 1 $ $3$ $3$ $(2,8,5)(3,6,9)$
$ 9 $ $3$ $9$ $(1,2,3,4,5,6,7,8,9)$
$ 9 $ $3$ $9$ $(1,2,6,4,5,9,7,8,3)$
$ 9 $ $3$ $9$ $(1,2,9,4,5,3,7,8,6)$
$ 9 $ $3$ $9$ $(1,3,8,7,9,5,4,6,2)$
$ 9 $ $3$ $9$ $(1,3,5,7,9,2,4,6,8)$
$ 9 $ $3$ $9$ $(1,3,2,7,9,8,4,6,5)$
$ 3, 3, 3 $ $1$ $3$ $(1,4,7)(2,5,8)(3,6,9)$
$ 3, 3, 3 $ $1$ $3$ $(1,7,4)(2,8,5)(3,9,6)$

Group invariants

Order:  $27=3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [27, 4]
Character table:   
      3  3  2  2  2  2  2  2  2  2  3  3

        1a 3a 3b 9a 9b 9c 9d 9e 9f 3c 3d
     2P 1a 3b 3a 9e 9d 9f 9b 9a 9c 3d 3c
     3P 1a 1a 1a 3c 3c 3c 3d 3d 3d 1a 1a
     5P 1a 3b 3a 9e 9d 9f 9b 9a 9c 3d 3c
     7P 1a 3a 3b 9a 9b 9c 9d 9e 9f 3c 3d

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