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Properties

Label 9T22
Order \(162\)
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 $t$ :  $22$
Group :  $(C_3^2:C_3):C_2$
CHM label :  $[3^{3}:2]3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,9), (1,2)(4,5)(7,8), (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:  $S_3\times C_3$
54:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

9T22 x 2, 18T85 x 3, 27T53 x 3, 27T62, 27T63

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 $ $6$ $3$ $(6,7,8)$
$ 3, 3, 1, 1, 1 $ $6$ $3$ $(3,4,5)(6,7,8)$
$ 3, 3, 1, 1, 1 $ $6$ $3$ $(3,4,5)(6,8,7)$
$ 2, 2, 2, 1, 1, 1 $ $27$ $2$ $(2,9)(4,5)(7,8)$
$ 3, 3, 3 $ $2$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 3, 3, 3 $ $6$ $3$ $(1,2,9)(3,4,5)(6,8,7)$
$ 3, 3, 3 $ $9$ $3$ $(1,3,6)(2,4,7)(5,8,9)$
$ 9 $ $18$ $9$ $(1,3,6,2,4,7,9,5,8)$
$ 6, 3 $ $27$ $6$ $(1,3,6)(2,5,7,9,4,8)$
$ 3, 3, 3 $ $9$ $3$ $(1,6,3)(2,7,4)(5,9,8)$
$ 9 $ $18$ $9$ $(1,6,4,2,7,5,9,8,3)$
$ 6, 3 $ $27$ $6$ $(1,6,3)(2,8,4,9,7,5)$

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 11]
Character table:    
      2  1  .  .  .  1  .  .  1   .   1  1   .   1
      3  4  3  3  3  1  4  3  2   2   1  2   2   1

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

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

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