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Properties

Label 6T3
Degree $6$
Order $12$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{6}$

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Show commands: Magma

magma: G := TransitiveGroup(6, 3);
 

Group action invariants

Degree $n$:  $6$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{6}$
CHM label:  $D(6) = S(3)[x]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4)(2,3)(5,6), (1,2,3,4,5,6)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Low degree siblings

6T3, 12T3

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $12=2^{2} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  12.4
magma: IdentifyGroup(G);
 
Character table:    
     2  2  2  2  1  1  2
     3  1  .  .  1  1  1

       1a 2a 2b 6a 3a 2c
    2P 1a 1a 1a 3a 3a 1a
    3P 1a 2a 2b 2c 1a 2c
    5P 1a 2a 2b 6a 3a 2c

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  .  .  1 -1 -2
X.6     2  .  . -1 -1  2

magma: CharacterTable(G);