# Properties

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

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## Group action invariants

 Degree $n$: $2$ Transitive number $t$: $1$ Group: $C_2$ CHM label: $S2$ Parity: $-1$ Primitive: yes Nilpotency class: $1$ $|\Aut(F/K)|$: $2$ Generators: (1,2)

## 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 Type Size Order Representative $1, 1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,2)$

## Group invariants

 Order: $2$ (is prime) Cyclic: yes Abelian: yes Solvable: yes GAP id: [2, 1]
 Character table:  2 1 1 1a 2a 2P 1a 1a X.1 1 -1 X.2 1 1 

## 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.