Properties

Label 9T23
Order \(216\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $(C_3^2:Q_8):C_3$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $23$
Group :  $(C_3^2:Q_8):C_3$
CHM label :  $E(9):2A_{4}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,9)(3,4,5)(6,7,8), (1,8,2,4)(3,5,6,7), (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)
3:  $C_3$
12:  $A_4$
24:  $\SL(2,3)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

12T122, 24T562, 24T569, 27T82, 36T287, 36T309

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 $ $12$ $3$ $(3,4,5)(6,8,7)$
$ 3, 3, 1, 1, 1 $ $12$ $3$ $(3,5,4)(6,7,8)$
$ 6, 2, 1 $ $36$ $6$ $(2,3,5,9,8,6)(4,7)$
$ 4, 4, 1 $ $54$ $4$ $(2,3,9,8)(4,5,7,6)$
$ 6, 2, 1 $ $36$ $6$ $(2,6,8,9,5,3)(4,7)$
$ 2, 2, 2, 2, 1 $ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$
$ 3, 3, 3 $ $24$ $3$ $(1,2,3)(4,5,6)(7,8,9)$
$ 3, 3, 3 $ $24$ $3$ $(1,2,6)(3,7,8)(4,5,9)$
$ 3, 3, 3 $ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$

Group invariants

Order:  $216=2^{3} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [216, 153]
Character table:   
      2  3   1   1  1  2  1  3   .   .  .
      3  3   2   2  1  .  1  1   2   2  3

        1a  3a  3b 6a 4a 6b 2a  3c  3d 3e
     2P 1a  3b  3a 3b 2a 3a 1a  3d  3c 3e
     3P 1a  1a  1a 2a 4a 2a 2a  1a  1a 1a
     5P 1a  3b  3a 6b 4a 6a 2a  3d  3c 3e

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

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