# Properties

 Label 4T2 Degree $4$ Order $4$ Cyclic no Abelian yes Solvable yes Primitive no $p$-group yes Group: $C_2^2$

# Related objects

## 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: $1$ $|\Aut(F/K)|$: $4$ Generators: (1,2)(3,4), (1,4)(2,3)

## 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 Type Size Order Representative $1, 1, 1, 1$ $1$ $1$ $()$ $2, 2$ $1$ $2$ $(1,2)(3,4)$ $2, 2$ $1$ $2$ $(1,3)(2,4)$ $2, 2$ $1$ $2$ $(1,4)(2,3)$

## 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.

This is the smallest transitive permutation group on $n$ elements which does not contain an $n$-cycle.