↑

Properties

Label 9T2
Degree $9$
Order $9$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_3^2$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(9, 2);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2$
CHM label:  $E(9)=3[x]3$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $9$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,9)(3,4,5)(6,7,8), (1,4,7)(2,5,8)(3,6,9)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Low degree siblings

There are no siblings 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 $ $1$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 3, 3, 3 $ $1$ $3$ $(1,3,8)(2,4,6)(5,7,9)$
$ 3, 3, 3 $ $1$ $3$ $(1,4,7)(2,5,8)(3,6,9)$
$ 3, 3, 3 $ $1$ $3$ $(1,5,6)(2,3,7)(4,8,9)$
$ 3, 3, 3 $ $1$ $3$ $(1,6,5)(2,7,3)(4,9,8)$
$ 3, 3, 3 $ $1$ $3$ $(1,7,4)(2,8,5)(3,9,6)$
$ 3, 3, 3 $ $1$ $3$ $(1,8,3)(2,6,4)(5,9,7)$
$ 3, 3, 3 $ $1$ $3$ $(1,9,2)(3,5,4)(6,8,7)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $9=3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  9.2
magma: IdentifyGroup(G);
 
Character table:    
     3  2  2  2  2  2  2  2  2  2

       1a 3a 3b 3c 3d 3e 3f 3g 3h
    2P 1a 3h 3g 3f 3e 3d 3c 3b 3a
    3P 1a 1a 1a 1a 1a 1a 1a 1a 1a

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

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

magma: CharacterTable(G);