# Properties

 Label 4T1 Order $4$ n $4$ Cyclic Yes Abelian Yes Solvable Yes Primitive No $p$-group Yes Group: $C_4$

# Related objects

## Group action invariants

Degree $n$ :  $4$
Transitive number $t$ :  $1$
Group :  $C_4$
CHM label :  $C(4) = 4$
Parity:  $-1$
Primitive:  No
Generators:   (1,2,3,4)
$|\Aut(F/K)|$:  $4$
Low degree resolvents:
 2: 2T1

## Subfields

Degree 2: $C_2$

## Low degree siblings

There is no other low degree representation.
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 $1$ $\left(\begin{array}{*{1}{r}}1\end{array}\right)$
$B$ $1$ $\left(\begin{array}{*{1}{r}}-1\end{array}\right)$
$C$ $2$ $\left(\begin{array}{*{2}{r}}0 & -1\\1 & 0\end{array}\right)$
$E$ $2$ $\left(\begin{array}{*{2}{r}}0 & 1\\1 & 0\end{array}\right)$
$(C,\text{Triv})$ $3$ $\left(\begin{array}{*{3}{r}}1 & 0 & 1\\0 & 0 & -1\\0 & 1 & 0\end{array}\right)$
$(C,B)$ $3$ $\left(\begin{array}{*{3}{r}}-1 & 0 & 1\\0 & 0 & -1\\0 & 1 & 0\end{array}\right)$
$(C,E)$ $4$ $\left(\begin{array}{*{4}{r}}0 & 1 & 0 & 0\\1 & 0 & 0 & 1\\0 & 0 & 0 & -1\\0 & 0 & 1 & 0\end{array}\right)$
$(C,E)'$ $4$ $\left(\begin{array}{*{4}{r}}0 & 1 & 0 & 1\\1 & 0 & 0 & -1\\0 & 0 & 0 & -1\\0 & 0 & 1 & 0\end{array}\right)$
$(C,\text{Triv}+B)$ $4$ $\left(\begin{array}{*{4}{r}}1 & 0 & 0 & 1\\0 & -1 & 0 & 1\\0 & 0 & 0 & -1\\0 & 0 & 1 & 0\end{array}\right)$
The decomposition of an arbitrary integral representation as a direct sum of indecomposables is unique.