# Properties

 Label 3T2 Degree $3$ Order $6$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $S_3$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(3, 2);

## Group action invariants

 Degree $n$: $3$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $2$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_3$ CHM label: $S3$ Parity: $-1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3), (1,2) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

6T2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Label Cycle Type Size Order Index Representative 1A $1^{3}$ $1$ $1$ $0$ $()$ 2A $2,1$ $3$ $2$ $1$ $(1,2)$ 3A $3$ $2$ $3$ $2$ $(1,2,3)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1$

## 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); Nilpotency class: not nilpotent Label: 6.1 magma: IdentifyGroup(G); Character table:

 1A 2A 3A Size 1 3 2 2 P 1A 1A 3A 3 P 1A 2A 1A Type 6.1.1a R $1$ $1$ $1$ 6.1.1b R $1$ $−1$ $1$ 6.1.2a R $2$ $0$ $−1$

magma: CharacterTable(G);

## Indecomposable integral representations

Complete list of indecomposable integral representations:

Name Dim $(1,2,3) \mapsto$ $(1,2) \mapsto$
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 $\oplus$ $(A+A',L)$ $\cong$ $(A,L)$ $\oplus$ $(A',L)$