# Properties

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

# Related objects

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 Type Size Order Representative $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1$ $3$ $2$ $(2,6)(3,5)$ $2, 2, 2$ $3$ $2$ $(1,2)(3,6)(4,5)$ $6$ $2$ $6$ $(1,2,3,4,5,6)$ $3, 3$ $2$ $3$ $(1,3,5)(2,4,6)$ $2, 2, 2$ $1$ $2$ $(1,4)(2,5)(3,6)$

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);