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Properties

Label 9T32
Degree $9$
Order $1512$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\mathrm{P}\Gamma\mathrm{L}(2,8)$

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Show commands: Magma

magma: G := TransitiveGroup(9, 32);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $32$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\mathrm{P}\Gamma\mathrm{L}(2,8)$
CHM label:  $L(9):3=P|L(2,8)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,5)(3,6)(4,7)(8,9), (1,9)(2,3)(4,5)(6,7), (1,2,4,3,6,7,5), (2,4,6)(3,5,7)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

27T391, 28T165, 36T2342

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 $ $1$ $1$ $()$
$ 7, 1, 1 $ $216$ $7$ $(2,4,7,6,5,3,8)$
$ 2, 2, 2, 2, 1 $ $63$ $2$ $(1,3)(2,7)(4,9)(5,8)$
$ 3, 3, 1, 1, 1 $ $84$ $3$ $(1,7,5)(2,8,3)$
$ 3, 3, 1, 1, 1 $ $84$ $3$ $(1,5,7)(2,3,8)$
$ 6, 2, 1 $ $252$ $6$ $(1,8,7,3,5,2)(4,9)$
$ 6, 2, 1 $ $252$ $6$ $(1,2,5,3,7,8)(4,9)$
$ 3, 3, 3 $ $56$ $3$ $(1,8,9)(2,6,4)(3,5,7)$
$ 9 $ $168$ $9$ $(1,6,3,8,4,5,9,2,7)$
$ 9 $ $168$ $9$ $(1,3,4,9,7,6,8,5,2)$
$ 9 $ $168$ $9$ $(1,5,4,9,3,6,8,7,2)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1512=2^{3} \cdot 3^{3} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1512.779
magma: IdentifyGroup(G);
 
Character table:    
      2  3  .  .   .   .  3  1  1   1   1  .
      3  3  3  2   2   2  1  2  2   1   1  .
      7  1  .  .   .   .  .  .  .   .   .  1

        1a 3a 9a  9b  9c 2a 3b 3c  6a  6b 7a
     2P 1a 3a 9a  9c  9b 1a 3c 3b  3b  3c 7a
     3P 1a 1a 3a  3a  3a 2a 1a 1a  2a  2a 7a
     5P 1a 3a 9a  9c  9b 2a 3c 3b  6b  6a 7a
     7P 1a 3a 9a  9b  9c 2a 3b 3c  6a  6b 1a

X.1      1  1  1   1   1  1  1  1   1   1  1
X.2      1  1  1   A  /A  1 /A  A   A  /A  1
X.3      1  1  1  /A   A  1  A /A  /A   A  1
X.4      7 -2  1   1   1 -1  1  1  -1  -1  .
X.5      7 -2  1   A  /A -1 /A  A  -A -/A  .
X.6      7 -2  1  /A   A -1  A /A -/A  -A  .
X.7      8 -1 -1  -1  -1  .  2  2   .   .  1
X.8      8 -1 -1 -/A  -A  .  B /B   .   .  1
X.9      8 -1 -1  -A -/A  . /B  B   .   .  1
X.10    21  3  .   .   . -3  .  .   .   .  .
X.11    27  .  .   .   .  3  .  .   .   . -1

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

magma: CharacterTable(G);