Properties

Label 9T28
Degree $9$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3 \wr C_3 $

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Show commands: Magma

magma: G := TransitiveGroup(9, 28);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $28$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3 \wr C_3 $
CHM label:  $[S(3)^{3}]3=S(3)wr3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,9), (1,2), (1,4,7)(2,5,8)(3,6,9)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

12T176, 18T197 x 2, 18T198 x 2, 18T202, 18T204, 18T206, 18T207, 24T1528, 24T1539, 27T210, 27T213, 36T1094 x 2, 36T1095, 36T1096 x 2, 36T1097, 36T1099, 36T1101, 36T1102, 36T1103, 36T1137, 36T1238

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, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $(1,2,9)$
$ 3, 3, 1, 1, 1 $ $12$ $3$ $(1,2,9)(3,4,5)$
$ 3, 3, 3 $ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 2, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $(2,9)$
$ 3, 2, 1, 1, 1, 1 $ $18$ $6$ $(2,9)(3,4,5)$
$ 3, 2, 1, 1, 1, 1 $ $18$ $6$ $(2,9)(6,7,8)$
$ 3, 3, 2, 1 $ $36$ $6$ $(2,9)(3,4,5)(6,7,8)$
$ 2, 2, 1, 1, 1, 1, 1 $ $27$ $2$ $(2,9)(4,5)$
$ 3, 2, 2, 1, 1 $ $54$ $6$ $(2,9)(4,5)(6,7,8)$
$ 2, 2, 2, 1, 1, 1 $ $27$ $2$ $(2,9)(4,5)(7,8)$
$ 3, 3, 3 $ $36$ $3$ $(1,4,7)(2,5,8)(3,6,9)$
$ 9 $ $72$ $9$ $(1,4,7,2,5,8,9,3,6)$
$ 6, 3 $ $108$ $6$ $(1,4,7)(2,5,8,9,3,6)$
$ 3, 3, 3 $ $36$ $3$ $(1,7,4)(2,8,5)(3,9,6)$
$ 9 $ $72$ $9$ $(1,7,4,2,8,5,9,6,3)$
$ 6, 3 $ $108$ $6$ $(1,7,4)(2,8,5,9,6,3)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $648=2^{3} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  648.705
magma: IdentifyGroup(G);
 
Character table:   
      2  3  2  1  .  3  2  2  1  3  2  3  1   .   1  1   .   1
      3  4  3  3  4  2  2  2  2  1  1  1  2   2   1  2   2   1

        1a 3a 3b 3c 2a 6a 6b 6c 2b 6d 2c 3d  9a  6e 3e  9b  6f
     2P 1a 3a 3b 3c 1a 3a 3a 3b 1a 3a 1a 3e  9b  3e 3d  9a  3d
     3P 1a 1a 1a 1a 2a 2a 2a 2a 2b 2b 2c 1a  3c  2c 1a  3c  2c
     5P 1a 3a 3b 3c 2a 6a 6b 6c 2b 6d 2c 3e  9b  6f 3d  9a  6e
     7P 1a 3a 3b 3c 2a 6a 6b 6c 2b 6d 2c 3d  9a  6e 3e  9b  6f

X.1      1  1  1  1  1  1  1  1  1  1  1  1   1   1  1   1   1
X.2      1  1  1  1 -1 -1 -1 -1  1  1 -1  1   1  -1  1   1  -1
X.3      1  1  1  1 -1 -1 -1 -1  1  1 -1  A   A  -A /A  /A -/A
X.4      1  1  1  1 -1 -1 -1 -1  1  1 -1 /A  /A -/A  A   A  -A
X.5      1  1  1  1  1  1  1  1  1  1  1  A   A   A /A  /A  /A
X.6      1  1  1  1  1  1  1  1  1  1  1 /A  /A  /A  A   A   A
X.7      3  3  3  3 -1 -1 -1 -1 -1 -1  3  .   .   .  .   .   .
X.8      3  3  3  3  1  1  1  1 -1 -1 -3  .   .   .  .   .   .
X.9      6  3  . -3  4  1  1 -2  2 -1  .  .   .   .  .   .   .
X.10     6  3  . -3 -4 -1 -1  2  2 -1  .  .   .   .  .   .   .
X.11     6  3  . -3  . -3  3  . -2  1  .  .   .   .  .   .   .
X.12     6  3  . -3  .  3 -3  . -2  1  .  .   .   .  .   .   .
X.13     8 -4  2 -1  .  .  .  .  .  .  .  2  -1   .  2  -1   .
X.14     8 -4  2 -1  .  .  .  .  .  .  .  B  -A   . /B -/A   .
X.15     8 -4  2 -1  .  .  .  .  .  .  . /B -/A   .  B  -A   .
X.16    12  . -3  3 -4  2  2 -1  .  .  .  .   .   .  .   .   .
X.17    12  . -3  3  4 -2 -2  1  .  .  .  .   .   .  .   .   .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3

magma: CharacterTable(G);