Properties

Label 9T1
Degree $9$
Order $9$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_9$

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Show commands: Magma

magma: G := TransitiveGroup(9, 1);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_9$
CHM label:  $C(9)=9$
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:  (1,2,3,4,5,6,7,8,9)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $9=3^{2}$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  9.1
magma: IdentifyGroup(G);
 
Character table:   
     3  2  2  2  2  2  2  2  2  2

       1a 9a 9b 3a 9c 9d 3b 9e 9f

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

A = E(3)
  = (-1+Sqrt(-3))/2 = b3
B = -E(9)^4-E(9)^7
C = E(9)^4
D = E(9)^7

magma: CharacterTable(G);