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Properties

Label 6T1
Degree $6$
Order $6$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_6$

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Show commands: Magma

magma: G := TransitiveGroup(6, 1);
 

Group action invariants

Degree $n$:  $6$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_6$
CHM label:  $C(6) = 6 = 3[x]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $1$
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5,6)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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 $ $1$ $1$ $()$
$ 6 $ $1$ $6$ $(1,2,3,4,5,6)$
$ 3, 3 $ $1$ $3$ $(1,3,5)(2,4,6)$
$ 2, 2, 2 $ $1$ $2$ $(1,4)(2,5)(3,6)$
$ 3, 3 $ $1$ $3$ $(1,5,3)(2,6,4)$
$ 6 $ $1$ $6$ $(1,6,5,4,3,2)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $6=2 \cdot 3$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  6.2
magma: IdentifyGroup(G);
 
Character table:    
     2  1   1  1  1  1   1
     3  1   1  1  1  1   1

       1a  6a 3a 2a 3b  6b

X.1     1   1  1  1  1   1
X.2     1  -1  1 -1  1  -1
X.3     1   A /A  1  A  /A
X.4     1  -A /A -1  A -/A
X.5     1  /A  A  1 /A   A
X.6     1 -/A  A -1 /A  -A

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

magma: CharacterTable(G);