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Properties

Label 9T12
Order \(54\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_3^2:C_3):C_2$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $12$
Group :  $(C_3^2:C_3):C_2$
CHM label :  $[3^{2}]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,6)(4,7)(5,8), (3,4,5)(6,8,7), (1,4,7)(2,5,8)(3,6,9)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

9T12 x 3, 18T24 x 4, 27T6

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

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 8]
Character table:    
      2  1  .  1  1   1   1  .  .  .  1
      3  3  2  1  3   1   1  2  2  2  3

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

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

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