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Properties

Label 9T13
Order \(54\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2 : S_3 $

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $13$
Group :  $C_3^2 : S_3 $
CHM label :  $E(9):D_{6}=[3^{2}:2]3=[1/2.S(3)^{2}]3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,5)(6,7), (1,2,9)(3,4,5)(6,7,8), (3,4,5)(6,8,7), (1,4,7)(2,5,8)(3,6,9)
$|\Aut(F/K)|$:  $1$

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: $C_3$

Low degree siblings

9T11, 18T20, 18T21, 18T22, 27T11

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

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 5]
Character table:    
      2  1  .  1  .   .  1   1   .  1   1
      3  3  2  1  3   2  2   1   2  2   1

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

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

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