Properties

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

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $32$
Group :  $\mathrm{P}\Gamma\mathrm{L}(2,8)$
CHM label :  $L(9):3=P|L(2,8)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

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

Group invariants

Order:  $1512=2^{3} \cdot 3^{3} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [1512, 779]
Character table:   
      2  3  3  1  1   1   1  .  .   .   .  .
      3  3  1  2  2   1   1  3  2   2   2  .
      7  1  .  .  .   .   .  .  .   .   .  1

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

X.1      1  1  1  1   1   1  1  1   1   1  1
X.2      1  1  A /A  /A   A  1  1  /A   A  1
X.3      1  1 /A  A   A  /A  1  1   A  /A  1
X.4      7 -1  1  1  -1  -1 -2  1   1   1  .
X.5      7 -1 /A  A  -A -/A -2  1   A  /A  .
X.6      7 -1  A /A -/A  -A -2  1  /A   A  .
X.7      8  .  2  2   .   . -1 -1  -1  -1  1
X.8      8  .  B /B   .   . -1 -1 -/A  -A  1
X.9      8  . /B  B   .   . -1 -1  -A -/A  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