Properties

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

Related objects

Downloads

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);
 
Nilpotency class:  $1$
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

Cycle TypeSizeOrderRepresentative
$ 1, 1 $ $1$ $1$ $()$
$ 2 $ $1$ $2$ $(1,2)$

magma: ConjugacyClasses(G);
 

Group invariants

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

       1a 2a
    2P 1a 1a

X.1     1 -1
X.2     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.