# Properties

 Label 3T1 Degree $3$ Order $3$ Cyclic yes Abelian yes Solvable yes Primitive yes $p$-group yes Group: $C_3$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(3, 1);

## Group action invariants

 Degree $n$: $3$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $1$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_3$ CHM label: $A3$ Parity: $1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $3$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,3) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## 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

 Label Cycle Type Size Order Index Representative 1A $1^{3}$ $1$ $1$ $0$ $()$ 3A1 $3$ $1$ $3$ $2$ $(1,2,3)$ 3A-1 $3$ $1$ $3$ $2$ $(1,3,2)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/2$

## Group invariants

 Order: $3$ (is prime) magma: Order(G); Cyclic: yes magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $1$ Label: 3.1 magma: IdentifyGroup(G); Character table:

 1A 3A1 3A-1 Size 1 1 1 3 P 1A 3A-1 3A1 Type 3.1.1a R $1$ $1$ $1$ 3.1.1b1 C $1$ $ζ3−1$ $ζ3$ 3.1.1b2 C $1$ $ζ3$ $ζ3−1$

magma: CharacterTable(G);

## Indecomposable integral representations

Complete list of indecomposable integral representations:

Name Dim $(1,2,3) \mapsto$
Triv $1$ $\left(\begin{array}{r}1\end{array}\right)$
$J$ $2$ $\left(\begin{array}{rr}0 & 1\\-1 & -1\end{array}\right)$
$R$ $3$ $\left(\begin{array}{rrr}0 & 1 & 0\\0 & 0 & 1\\1 & 0 & 0\end{array}\right)$
The decomposition of an arbitrary integral representation as a direct sum of indecomposables is unique.