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Properties

Label 9T14
Degree $9$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_3^2:Q_8$

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Show commands: Magma

magma: G := TransitiveGroup(9, 14);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $14$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2:Q_8$
CHM label:  $M(9)=E(9):Q_{8}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  72.41
magma: IdentifyGroup(G);
 
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

magma: CharacterTable(G);