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 TypeSizeOrderRepresentative
$ 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.