Properties

Label 9T19
Order \(144\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $(C_3^2:C_8):C_2$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $19$
Group :  $(C_3^2:C_8):C_2$
CHM label :  $E(9):2D_{8}$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,4,5,2,3,8,7), (1,2)(3,5)(6,7), (1,2,9)(3,4,5)(6,7,8), (1,4,7)(2,5,8)(3,6,9)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
16:  $QD_{16}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

12T84, 18T68, 18T71, 18T73, 24T278, 24T279, 24T280, 36T171, 36T172, 36T175

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 $ $1$ $1$ $()$
$ 2, 2, 2, 1, 1, 1 $ $12$ $2$ $(3,7)(4,8)(5,6)$
$ 8, 1 $ $18$ $8$ $(2,3,6,7,9,8,5,4)$
$ 4, 4, 1 $ $36$ $4$ $(2,3,9,8)(4,5,7,6)$
$ 8, 1 $ $18$ $8$ $(2,4,5,8,9,7,6,3)$
$ 4, 4, 1 $ $18$ $4$ $(2,5,9,6)(3,4,8,7)$
$ 2, 2, 2, 2, 1 $ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$
$ 6, 3 $ $24$ $6$ $(1,2,3,4,8,6)(5,9,7)$
$ 3, 3, 3 $ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$

Group invariants

Order:  $144=2^{4} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [144, 182]
Character table:   
     2  4  2  3  2  3  3  4  1  1
     3  2  1  .  .  .  .  .  1  2

       1a 2a 8a 4a 8b 4b 2b 6a 3a
    2P 1a 1a 4b 2b 4b 2b 1a 3a 3a
    3P 1a 2a 8a 4a 8b 4b 2b 2a 1a
    5P 1a 2a 8b 4a 8a 4b 2b 6a 3a
    7P 1a 2a 8b 4a 8a 4b 2b 6a 3a

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

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2