Properties

Label 9T24
Order \(324\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $((C_3^3:C_3):C_2):C_2$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $24$
Group :  $((C_3^3:C_3):C_2):C_2$
CHM label :  $[3^{3}:2]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,9), (3,6)(4,7)(5,8), (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:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
12:  $D_{6}$ x 2
36:  $S_3^2$
108:  $C_3^2 : D_{6} $

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

9T24 x 2, 18T129 x 3, 18T136 x 3, 18T137 x 3, 27T121, 27T128 x 3, 27T129, 36T502 x 3

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 $ $9$ $2$ $(3,6)(4,7)(5,8)$
$ 6, 1, 1, 1 $ $18$ $6$ $(3,6,4,7,5,8)$
$ 2, 2, 2, 1, 1, 1 $ $27$ $2$ $(2,9)(4,5)(7,8)$
$ 2, 2, 2, 2, 1 $ $27$ $2$ $(2,9)(3,6)(4,8)(5,7)$
$ 6, 2, 1 $ $54$ $6$ $(2,9)(3,6,4,8,5,7)$
$ 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, 2, 2, 2 $ $18$ $6$ $(1,2,9)(3,6)(4,7)(5,8)$
$ 6, 3 $ $18$ $6$ $(1,2,9)(3,6,4,7,5,8)$
$ 6, 3 $ $18$ $6$ $(1,2,9)(3,6,5,8,4,7)$
$ 3, 3, 3 $ $18$ $3$ $(1,3,6)(2,4,7)(5,8,9)$
$ 9 $ $36$ $9$ $(1,3,6,2,4,7,9,5,8)$
$ 6, 3 $ $54$ $6$ $(1,3,6)(2,5,7,9,4,8)$

Group invariants

Order:  $324=2^{2} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [324, 39]
Character table:   
      2  2  1  1  1  2  1  2  2  1  1  1  1  1  1  1  .  1
      3  4  3  3  3  2  2  1  1  1  4  3  2  2  2  2  2  1

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

X.1      1  1  1  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 -1  1  1 -1
X.3      1  1  1  1 -1 -1  1 -1 -1  1  1 -1 -1 -1  1  1  1
X.4      1  1  1  1  1  1 -1 -1 -1  1  1  1  1  1  1  1 -1
X.5      2  2  2  2  .  . -2  .  .  2  2  .  .  . -1 -1  1
X.6      2  2  2  2  .  .  2  .  .  2  2  .  .  . -1 -1 -1
X.7      2 -1 -1  2 -2  1  .  .  .  2 -1  1  1 -2  2 -1  .
X.8      2 -1 -1  2  2 -1  .  .  .  2 -1 -1 -1  2  2 -1  .
X.9      4 -2 -2  4  .  .  .  .  .  4 -2  .  .  . -2  1  .
X.10     6 -3  3  . -2  1  .  .  . -3  .  1 -2  1  .  .  .
X.11     6 -3  3  .  2 -1  .  .  . -3  . -1  2 -1  .  .  .
X.12     6  . -3  . -2  1  .  .  . -3  3 -2  1  1  .  .  .
X.13     6  . -3  .  2 -1  .  .  . -3  3  2 -1 -1  .  .  .
X.14     6  .  . -3  .  .  . -2  1  6  .  .  .  .  .  .  .
X.15     6  .  . -3  .  .  .  2 -1  6  .  .  .  .  .  .  .
X.16     6  3  .  . -2 -2  .  .  . -3 -3  1  1  1  .  .  .
X.17     6  3  .  .  2  2  .  .  . -3 -3 -1 -1 -1  .  .  .