Properties

Label 9T6
Degree $9$
Order $27$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_9:C_3$

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Show commands: Magma

magma: G := TransitiveGroup(9, 6);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_9:C_3$
CHM label:  $1/3[3^{3}]3$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4,7)(2,8,5), (1,2,3,4,5,6,7,8,9)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $27=3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  27.4
magma: IdentifyGroup(G);
 
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

magma: CharacterTable(G);