# Properties

 Label 6T2 Degree $6$ Order $6$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(6, 2);

## Group action invariants

 Degree $n$: $6$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $2$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_3$ CHM label: $D_{6}(6) = [3]2$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $6$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3,5)(2,4,6), (1,4)(2,3)(5,6) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

## Low degree siblings

3T2

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $6=2 \cdot 3$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 6.1 magma: IdentifyGroup(G);
 Character table:  2 1 1 . 3 1 . 1 1a 2a 3a 2P 1a 1a 3a 3P 1a 2a 1a X.1 1 1 1 X.2 1 -1 1 X.3 2 . -1 

magma: CharacterTable(G);

## Indecomposable integral representations

Complete list of indecomposable integral representations:

Name Dim $(1,3,5)(2,4,6) \mapsto$ $(1,2)(3,6)(4,5) \mapsto$
Triv $1$ $\left(\begin{array}{r}1\end{array}\right)$ $\left(\begin{array}{r}1\end{array}\right)$
Sign $1$ $\left(\begin{array}{r}1\end{array}\right)$ $\left(\begin{array}{r}-1\end{array}\right)$
$L$ $2$ $\left(\begin{array}{rr}1 & 0\\0 & 1\end{array}\right)$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$
$A$ $2$ $\left(\begin{array}{rr}0 & 1\\-1 & -1\end{array}\right)$ $\left(\begin{array}{rr}1 & 0\\-1 & -1\end{array}\right)$
$A'$ $2$ $\left(\begin{array}{rr}0 & 1\\-1 & -1\end{array}\right)$ $\left(\begin{array}{rr}-1 & 0\\1 & 1\end{array}\right)$
$(A,\textrm{Sign})$ $3$ $\left(\begin{array}{rrr}0 & 1 & 0\\-1 & -1 & 0\\1 & 0 & 1\end{array}\right)$ $\left(\begin{array}{rrr}1 & 0 & 0\\-1 & -1 & 0\\-1 & 0 & -1\end{array}\right)$
$(A',\textrm{Triv})$ $3$ $\left(\begin{array}{rrr}0 & 1 & 0\\-1 & -1 & 0\\1 & 0 & 1\end{array}\right)$ $\left(\begin{array}{rrr}-1 & 0 & 0\\1 & 1 & 0\\1 & 0 & 1\end{array}\right)$
$(A,L)$ $4$ $\left(\begin{array}{rrrr}0 & 1 & 0 & 0\\-1 & -1 & 0 & 0\\-1 & 0 & 1 & 0\\1 & 0 & 0 & 1\end{array}\right)$ $\left(\begin{array}{rrrr}1 & 0 & 0 & 0\\-1 & -1 & 0 & 0\\1 & 0 & 0 & 1\\-1 & 0 & 1 & 0\end{array}\right)$
$(A',L)$ $4$ $\left(\begin{array}{rrrr}0 & 1 & 0 & 0\\-1 & -1 & 0 & 0\\1 & 0 & 1 & 0\\1 & 0 & 0 & 1\end{array}\right)$ $\left(\begin{array}{rrrr}-1 & 0 & 0 & 0\\1 & 1 & 0 & 0\\1 & 0 & 0 & 1\\1 & 0 & 1 & 0\end{array}\right)$
$(A+A',L)$ $6$ $\left(\begin{array}{rrrrrr}0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 1\\1 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0\end{array}\right)$ $\left(\begin{array}{rrrrrr}0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0\\1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 0 & 0 & 1 & 0\end{array}\right)$
The decomposition of an arbitrary integral representation as a direct sum of indecomposables is not unique, in general. It is unique up to the following isomorphisms:
 Triv $\oplus$ $(A',L)$ $\cong$ $L$ $\oplus$ $(A',\textrm{Triv})$ Sign $\oplus$ $(A,L)$ $\cong$ $L$ $\oplus$ $(A,\textrm{Sign})$ Triv $\oplus$ $(A+A',L)$ $\cong$ $(A,L)$ $\oplus$ $(A',L)$