Properties

Label 9T25
Order \(324\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $25$
Group :  $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$
CHM label :  $[1/2.S(3)^{3}]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,9), (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)
3:  $C_3$
12:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

12T132 x 2, 12T133, 18T141 x 2, 18T142, 18T143, 27T130, 27T131, 36T511, 36T512, 36T524, 36T546 x 2, 36T547

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 $ $6$ $3$ $(6,7,8)$
$ 2, 2, 1, 1, 1, 1, 1 $ $27$ $2$ $(4,5)(7,8)$
$ 3, 3, 1, 1, 1 $ $12$ $3$ $(3,4,5)(6,7,8)$
$ 3, 2, 2, 1, 1 $ $54$ $6$ $(2,9)(4,5)(6,7,8)$
$ 3, 3, 3 $ $4$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 3, 3, 3 $ $4$ $3$ $(1,2,9)(3,4,5)(6,8,7)$
$ 3, 3, 3 $ $36$ $3$ $(1,3,6)(2,4,7)(5,8,9)$
$ 9 $ $36$ $9$ $(1,3,6,2,4,7,9,5,8)$
$ 9 $ $36$ $9$ $(1,3,6,9,5,8,2,4,7)$
$ 3, 3, 3 $ $36$ $3$ $(1,6,3)(2,7,4)(5,9,8)$
$ 9 $ $36$ $9$ $(1,6,4,2,7,5,9,8,3)$
$ 9 $ $36$ $9$ $(1,6,5,9,8,4,2,7,3)$

Group invariants

Order:  $324=2^{2} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [324, 160]
Character table:   
      2  2  1  2  .  1  .  .  .  .  .  .  .  .
      3  4  3  1  3  1  4  4  2  2  2  2  2  2

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

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

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