Properties

Label 9T11
Order \(54\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2 : C_6$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $11$
Group :  $C_3^2 : C_6$
CHM label :  $E(9):6=1/2[3^{2}:2]S(3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,9)(3,4,5)(6,7,8), (1,2)(3,6)(4,8)(5,7), (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: $S_3$

Low degree siblings

9T13, 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 $ $3$ $3$ $(3,4,5)(6,8,7)$
$ 3, 3, 1, 1, 1 $ $3$ $3$ $(3,5,4)(6,7,8)$
$ 6, 2, 1 $ $9$ $6$ $(2,9)(3,6,4,8,5,7)$
$ 6, 2, 1 $ $9$ $6$ $(2,9)(3,7,5,8,4,6)$
$ 2, 2, 2, 2, 1 $ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$
$ 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 $ $6$ $3$ $(1,3,7)(2,4,8)(5,6,9)$
$ 3, 3, 3 $ $6$ $3$ $(1,3,8)(2,4,6)(5,7,9)$

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  2   1   1  1  3   2   2  2

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

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

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