Properties

Label 9T30
Order \(648\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $30$
Group :  $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
CHM label :  $1/2[S(3)^{3}]S(3)$
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), (1,2)(3,6)(4,7)(5,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

12T177 x 2, 12T178, 18T217, 18T218, 18T221, 18T222, 24T1529 x 2, 24T1530, 27T211, 27T216, 36T1121, 36T1122, 36T1123, 36T1124, 36T1125, 36T1130, 36T1140, 36T1239 x 2, 36T1240

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 $ $6$ $3$ $(1,2,9)$
$ 3, 3, 1, 1, 1 $ $12$ $3$ $(1,2,9)(3,4,5)$
$ 3, 3, 3 $ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 2, 2, 1, 1, 1, 1, 1 $ $27$ $2$ $(4,5)(7,8)$
$ 3, 2, 2, 1, 1 $ $54$ $6$ $(1,2,9)(4,5)(7,8)$
$ 3, 3, 3 $ $72$ $3$ $(1,4,7)(2,5,8)(3,6,9)$
$ 9 $ $72$ $9$ $(1,4,7,2,5,8,9,3,6)$
$ 9 $ $72$ $9$ $(1,4,7,9,3,6,2,5,8)$
$ 2, 2, 2, 2, 1 $ $54$ $2$ $(2,9)(3,6)(4,7)(5,8)$
$ 6, 2, 1 $ $108$ $6$ $(2,9)(3,6,4,7,5,8)$
$ 4, 2, 1, 1, 1 $ $54$ $4$ $(3,6)(4,8,5,7)$
$ 4, 3, 2 $ $54$ $12$ $(1,2,9)(3,6)(4,8,5,7)$
$ 4, 3, 2 $ $54$ $12$ $(1,9,2)(3,6)(4,8,5,7)$

Group invariants

Order:  $648=2^{3} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [648, 703]
Character table:   
      2  3  2  1  .  3  2  .  .  .  2  1  2   2   2
      3  4  3  3  4  1  1  2  2  2  1  1  1   1   1

        1a 3a 3b 3c 2a 6a 3d 9a 9b 2b 6b 4a 12a 12b
     2P 1a 3a 3b 3c 1a 3a 3d 9a 9b 1a 3b 2a  6a  6a
     3P 1a 1a 1a 1a 2a 2a 1a 3c 3c 2b 2b 4a  4a  4a
     5P 1a 3a 3b 3c 2a 6a 3d 9a 9b 2b 6b 4a 12b 12a
     7P 1a 3a 3b 3c 2a 6a 3d 9a 9b 2b 6b 4a 12b 12a
    11P 1a 3a 3b 3c 2a 6a 3d 9a 9b 2b 6b 4a 12a 12b

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

A = -E(12)^7+E(12)^11
  = Sqrt(3) = r3