Properties

Label 4T2
4T2 1 2 1->2 4 1->4 3 2->3 3->4
Degree $4$
Order $4$
Cyclic no
Abelian yes
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2^2$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(4, 2);
 
Copy content sage:G = TransitiveGroup(4, 2)
 
Copy content oscar:G = transitive_group(4, 2)
 
Copy content gap:G := TransitiveGroup(4, 2);
 

Group invariants

Abstract group:  $C_2^2$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $4=2^{2}$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  yes
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:  $1$
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $4$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $2$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
CHM label:   $E(4) = 2[x]2$
Parity:  $1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(4).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(4), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(4), G));
 
Generators:  $(1,2)(3,4)$, $(1,4)(2,3)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$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

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{4}$ $1$ $1$ $0$ $()$
2A $2^{2}$ $1$ $2$ $2$ $(1,2)(3,4)$
2B $2^{2}$ $1$ $2$ $2$ $(1,4)(2,3)$
2C $2^{2}$ $1$ $2$ $2$ $(1,3)(2,4)$

Malle's constant $a(G)$:     $1/2$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C
Size 1 1 1 1
2 P 1A 1A 1A 1A
Type
4.2.1a R 1 1 1 1
4.2.1b R 1 −1 −1 1
4.2.1c R 1 −1 1 −1
4.2.1d R 1 1 −1 −1

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap: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.

Regular extensions

$f_{ 1 } =$ $x^{4}+s x^{2}+t^{2}$ Copy content Toggle raw display
The polynomial $f_{1}$ is generic for any base field $K$ of characteristic $\neq$ 2

Additional information

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