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Properties

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

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Show commands: Magma

magma: G := TransitiveGroup(4, 2);
 

Group action invariants

Degree $n$:  $4$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2$
CHM label:  $E(4) = 2[x]2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $1$
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2)(3,4), (1,4)(2,3)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $4=2^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  4.2
magma: IdentifyGroup(G);
 
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

magma: CharacterTable(G);
 

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.

Additional information

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