Galois group: 2T1

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

Group action invariants

Degree $n$:		$2$
Transitive number $t$:		$1$
Group:		$C_2$
CHM label:		$S2$
Parity:		$-1$
Primitive:		yes
Nilpotency class:		$1$
$\card{\Aut(F/K)}$:		$2$
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
Label:		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.