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Properties

Label 2T1
2T1 1 2 1->2
Degree $2$
Order $2$
Cyclic yes
Abelian yes
Solvable yes
Primitive yes
$p$-group yes
Group: $C_2$

Related objects

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Show commands: Magma

Copy content magma:G := TransitiveGroup(2, 1);
 

Group invariants

Abstract group:  $C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $2$ (is prime)
Copy content magma:Order(G);
 
Cyclic:  yes
Copy content magma:IsCyclic(G);
 
Abelian:  yes
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $1$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

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

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{2}$ $1$ $1$ $0$ $()$
2A $2$ $1$ $2$ $1$ $(1,2)$

Malle's constant $a(G)$:     $1$

Copy content magma:ConjugacyClasses(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

Copy content 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.

Regular extensions

Data not computed