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Properties

Label 9T29
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 $t$ :  $29$
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), (3,6)(4,7)(5,8), (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:  $C_2$
6:  $S_3$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

12T175, 18T219, 18T220, 18T223, 18T224, 24T1527, 24T1540, 27T214, 27T217, 36T1126, 36T1127, 36T1128, 36T1129, 36T1131, 36T1132, 36T1139, 36T1237

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 $ $4$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 3, 3, 3 $ $4$ $3$ $(1,9,2)(3,4,5)(6,7,8)$
$ 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 $ $72$ $3$ $(1,7,4)(2,8,5)(3,9,6)$
$ 9 $ $72$ $9$ $(1,7,4,2,8,5,9,6,3)$
$ 9 $ $72$ $9$ $(1,7,4,9,6,3,2,8,5)$
$ 2, 2, 2, 1, 1, 1 $ $18$ $2$ $(3,6)(4,7)(5,8)$
$ 3, 2, 2, 2 $ $18$ $6$ $(1,2,9)(3,6)(4,7)(5,8)$
$ 3, 2, 2, 2 $ $18$ $6$ $(1,9,2)(3,6)(4,7)(5,8)$
$ 6, 1, 1, 1 $ $36$ $6$ $(3,6,4,7,5,8)$
$ 6, 3 $ $36$ $6$ $(1,2,9)(3,6,4,7,5,8)$
$ 6, 3 $ $36$ $6$ $(1,9,2)(3,6,4,7,5,8)$
$ 4, 2, 2, 1 $ $162$ $4$ $(2,9)(3,6)(4,8,5,7)$

Group invariants

Order:  $648=2^{3} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [648, 704]
Character table:    
      2  3  2  1  1  1  3  2  .   .   .  2   2   2  1   1   1  2
      3  4  3  3  4  4  1  1  2   2   2  2   2   2  2   2   2  .

        1a 3a 3b 3c 3d 2a 6a 3e  9a  9b 2b  6b  6c 6d  6e  6f 4a
     2P 1a 3a 3b 3d 3c 1a 3a 3e  9b  9a 1a  3a  3a 3b  3d  3c 2a
     3P 1a 1a 1a 1a 1a 2a 2a 1a  3c  3d 2b  2b  2b 2b  2b  2b 4a
     5P 1a 3a 3b 3d 3c 2a 6a 3e  9b  9a 2b  6c  6b 6d  6f  6e 4a
     7P 1a 3a 3b 3c 3d 2a 6a 3e  9a  9b 2b  6b  6c 6d  6e  6f 4a

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      2  2  2  2  2  2  2 -1  -1  -1  .   .   .  .   .   .  .
X.4      3  3  3  3  3 -1 -1  .   .   .  1   1   1  1   1   1 -1
X.5      3  3  3  3  3 -1 -1  .   .   . -1  -1  -1 -1  -1  -1  1
X.6      4 -2  1  A /A  .  .  1   C  /C -2   D  /D  1   C  /C  .
X.7      4 -2  1 /A  A  .  .  1  /C   C -2  /D   D  1  /C   C  .
X.8      4 -2  1  A /A  .  .  1   C  /C  2  -D -/D -1  -C -/C  .
X.9      4 -2  1 /A  A  .  .  1  /C   C  2 -/D  -D -1 -/C  -C  .
X.10     6  3  . -3 -3  2 -1  .   .   .  2  -1  -1  2  -1  -1  .
X.11     6  3  . -3 -3  2 -1  .   .   . -2   1   1 -2   1   1  .
X.12     6  3  . -3 -3 -2  1  .   .   .  .   E  -E  .   E  -E  .
X.13     6  3  . -3 -3 -2  1  .   .   .  .  -E   E  .  -E   E  .
X.14     8 -4  2  B /B  .  . -1  -C -/C  .   .   .  .   .   .  .
X.15     8 -4  2 /B  B  .  . -1 -/C  -C  .   .   .  .   .   .  .
X.16    12  . -3  3  3  .  .  .   .   . -2  -2  -2  1   1   1  .
X.17    12  . -3  3  3  .  .  .   .   .  2   2   2 -1  -1  -1  .

A = -E(3)+2*E(3)^2
  = (-1-3*Sqrt(-3))/2 = -2-3b3
B = -2*E(3)+4*E(3)^2
  = -1-3*Sqrt(-3) = -1-3i3
C = E(3)
  = (-1+Sqrt(-3))/2 = b3
D = -2*E(3)
  = 1-Sqrt(-3) = 1-i3
E = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3