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Properties

Label 4T2
Order \(4\)
n \(4\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $4$
Transitive number $t$ :  $2$
Group :  $C_2^2$
CHM label :  $E(4) = 2[x]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,2)(3,4), (1,4)(2,3)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Group invariants

Order:  $4=2^{2}$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [4, 2]
Character table:    
     2  2  2  2  2

       1a 2a 2b 2c
    2P 1a 1a 1a 1a

X.1     1  1  1  1
X.2     1 -1 -1  1
X.3     1 -1  1 -1
X.4     1  1 -1 -1

Indecomposable integral representations

Partial list of indecomposable integral representations:

Name Dim $(1,2)(3,4) \mapsto $ $(1,4)(2,3) \mapsto $
Triv $1$ $\left(\begin{array}{r}1\end{array}\right)$ $\left(\begin{array}{r}1\end{array}\right)$
$A_1$ $1$ $\left(\begin{array}{r}1\end{array}\right)$ $\left(\begin{array}{r}-1\end{array}\right)$
$A_2$ $1$ $\left(\begin{array}{r}-1\end{array}\right)$ $\left(\begin{array}{r}1\end{array}\right)$
$A_3$ $1$ $\left(\begin{array}{r}-1\end{array}\right)$ $\left(\begin{array}{r}-1\end{array}\right)$
$B_1$ $2$ $\left(\begin{array}{rr}-1 & 0\\0 & -1\end{array}\right)$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$
$B_2$ $2$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $\left(\begin{array}{rr}-1 & 0\\0 & -1\end{array}\right)$
$B_3$ $2$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $\left(\begin{array}{rr}0 & -1\\-1 & 0\end{array}\right)$
$A_2B_1$ $2$ $\left(\begin{array}{rr}1 & 0\\0 & 1\end{array}\right)$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$
$A_3B_1$ $2$ $\left(\begin{array}{rr}1 & 0\\0 & 1\end{array}\right)$ $\left(\begin{array}{rr}0 & -1\\-1 & 0\end{array}\right)$
$A_1B_2$ $2$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $\left(\begin{array}{rr}1 & 0\\0 & 1\end{array}\right)$
$A_3B_2$ $2$ $\left(\begin{array}{rr}0 & -1\\-1 & 0\end{array}\right)$ $\left(\begin{array}{rr}1 & 0\\0 & 1\end{array}\right)$
$A_1B_3$ $2$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$
$A_2B_3$ $2$ $\left(\begin{array}{rr}0 & -1\\-1 & 0\end{array}\right)$ $\left(\begin{array}{rr}0 & -1\\-1 & 0\end{array}\right)$
$J$ $3$ $\left(\begin{array}{rrr}0 & -1 & 1\\0 & -1 & 0\\1 & -1 & 0\end{array}\right)$ $\left(\begin{array}{rrr}0 & 1 & -1\\1 & 0 & -1\\0 & 0 & -1\end{array}\right)$
$J'$ $3$ $\left(\begin{array}{rrr}0 & 0 & 1\\-1 & -1 & -1\\1 & 0 & 0\end{array}\right)$ $\left(\begin{array}{rrr}0 & 1 & 0\\1 & 0 & 0\\-1 & -1 & -1\end{array}\right)$
The decomposition of an arbitrary integral representation as a direct sum of indecomposables is unique.