# Properties

 Label 2T1 Degree $2$ Order $2$ Cyclic yes Abelian yes Solvable yes Primitive yes $p$-group yes Group: $C_2$

# Learn more

Show commands: Magma

magma: G := TransitiveGroup(2, 1);

## Group action invariants

 Degree $n$: $2$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $1$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_2$ CHM label: $S2$ Parity: $-1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2) 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 $1^{2}$ $1$ $1$ $0$ $()$ 2A $2$ $1$ $2$ $1$ $(1,2)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1$

## Group invariants

 Order: $2$ (is prime) magma: Order(G); Cyclic: yes magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $1$ Label: 2.1 magma: IdentifyGroup(G); Character table:

 1A 2A Size 1 1 2 P 1A 1A Type 2.1.1a R $1$ $1$ 2.1.1b R $1$ $−1$

magma: CharacterTable(G);

## Indecomposable integral representations

Complete list of indecomposable integral representations:

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