# Properties

 Label 4T1 Degree $4$ Order $4$ Cyclic yes Abelian yes Solvable yes Primitive no $p$-group yes Group: $C_4$

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## Group action invariants

 Degree $n$: $4$ Transitive number $t$: $1$ Group: $C_4$ CHM label: $C(4) = 4$ Parity: $-1$ Primitive: no Nilpotency class: $1$ $|\Aut(F/K)|$: $4$ Generators: (1,2,3,4)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

## 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 $1, 1, 1, 1$ $1$ $1$ $()$ $4$ $1$ $4$ $(1,2,3,4)$ $2, 2$ $1$ $2$ $(1,3)(2,4)$ $4$ $1$ $4$ $(1,4,3,2)$

## Group invariants

 Order: $4=2^{2}$ Cyclic: yes Abelian: yes Solvable: yes GAP id: [4, 1]
 Character table:  2 2 2 2 2 1a 4a 2a 4b X.1 1 1 1 1 X.2 1 -1 1 -1 X.3 1 A -1 -A X.4 1 -A -1 A A = E(4) = Sqrt(-1) = i 

## Indecomposable integral representations

Complete list of indecomposable integral representations:

Name Dim $(1,2,3,4) \mapsto$
Triv $1$ $\left(\begin{array}{r}1\end{array}\right)$
$B$ $1$ $\left(\begin{array}{r}-1\end{array}\right)$
$C$ $2$ $\left(\begin{array}{rr}0 & -1\\1 & 0\end{array}\right)$
$E$ $2$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$
$(C,\text{Triv})$ $3$ $\left(\begin{array}{rrr}1 & 0 & 1\\0 & 0 & -1\\0 & 1 & 0\end{array}\right)$
$(C,B)$ $3$ $\left(\begin{array}{rrr}-1 & 0 & 1\\0 & 0 & -1\\0 & 1 & 0\end{array}\right)$
$(C,E)$ $4$ $\left(\begin{array}{rrrr}0 & 1 & 0 & 0\\1 & 0 & 0 & 1\\0 & 0 & 0 & -1\\0 & 0 & 1 & 0\end{array}\right)$
$(C,E)'$ $4$ $\left(\begin{array}{rrrr}0 & 1 & 0 & 1\\1 & 0 & 0 & -1\\0 & 0 & 0 & -1\\0 & 0 & 1 & 0\end{array}\right)$
$(C,\text{Triv}+B)$ $4$ $\left(\begin{array}{rrrr}1 & 0 & 0 & 1\\0 & -1 & 0 & 1\\0 & 0 & 0 & -1\\0 & 0 & 1 & 0\end{array}\right)$
The decomposition of an arbitrary integral representation as a direct sum of indecomposables is unique.