Properties

Label 9T14
Order \(72\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_3^2:Q_8$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $14$
Group :  $C_3^2:Q_8$
CHM label :  $M(9)=E(9):Q_{8}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,9)(3,4,5)(6,7,8), (1,8,2,4)(3,5,6,7), (1,6,2,3)(4,7,8,5), (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:  $Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

12T47, 18T35 x 3, 24T82, 36T55

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 41]
Character table:   
     2  3  2  2  2  3  .
     3  2  .  .  .  .  2

       1a 4a 4b 4c 2a 3a
    2P 1a 2a 2a 2a 1a 3a
    3P 1a 4a 4b 4c 2a 1a

X.1     1  1  1  1  1  1
X.2     1 -1 -1  1  1  1
X.3     1 -1  1 -1  1  1
X.4     1  1 -1 -1  1  1
X.5     2  .  .  . -2  2
X.6     8  .  .  .  . -1