Properties

 Label 2T1 Order $$2$$ n $$2$$ 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 Generators: (1,2) $|\Aut(F/K)|$: $2$

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.