Properties

Label 4T4
4T4 1 3 1->3 2 2->3 4 3->4 3->4 4->1 4->2
Degree $4$
Order $12$
Cyclic no
Abelian no
Solvable yes
Transitivity $2$
Primitive yes
$p$-group no
Group: $A_4$

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Show commands: Gap / Magma / Oscar / SageMath

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

Group invariants

Abstract group:  $A_4$
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:  $12=2^{2} \cdot 3$
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:  no
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:   not nilpotent
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 $t$:  $4$
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:   $A4$
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:  2
Primitive:  yes
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);
 
$\card{\Aut(F/K)}$:  $1$
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:  $(2,3,4)$, $(1,3,4)$
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)
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Low degree siblings

6T4, 12T4

Siblings are shown 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 $2^{2}$ $3$ $2$ $2$ $(1,2)(3,4)$
3A1 $3,1$ $4$ $3$ $2$ $(2,4,3)$
3A-1 $3,1$ $4$ $3$ $2$ $(2,3,4)$

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 3A1 3A-1
Size 1 3 4 4
2 P 1A 1A 3A-1 3A1
3 P 1A 2A 1A 1A
Type
12.3.1a R 1 1 1 1
12.3.1b1 C 1 1 ζ3−1 ζ3
12.3.1b2 C 1 1 ζ3 ζ3−1
12.3.3a R 3 −1 0 0

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);
 

Regular extensions

$f_{ 1 } =$ $x^{4}+\left(\left(-6 s^{3}+\left(6 t^{3}+54 t^{2}+162 t+324\right)\right)/\left(s^{3}+\left(-3 t^{2}-9 t-27\right) s+\left(2 t^{3}+9 t^{2}+27 t+27\right)\right)\right) x^{2}-8 x+\left(\left(-3 s^{6}+\left(36 t^{2}+108 t+324\right) s^{4}+\left(-30 t^{3}-270 t^{2}-810 t-1620\right) s^{3}+\left(-36 t^{5}-108 t^{4}-324 t^{3}+972 t^{2}+2916 t+8748\right) s+\left(33 t^{6}+270 t^{5}+1539 t^{4}+5022 t^{3}+12393 t^{2}+17496 t+17496\right)\right)/\left(s^{6}+\left(-6 t^{2}-18 t-54\right) s^{4}+\left(4 t^{3}+18 t^{2}+54 t+54\right) s^{3}+\left(9 t^{4}+54 t^{3}+243 t^{2}+486 t+729\right) s^{2}+\left(-12 t^{5}-90 t^{4}-432 t^{3}-1134 t^{2}-1944 t-1458\right) s+\left(4 t^{6}+36 t^{5}+189 t^{4}+594 t^{3}+1215 t^{2}+1458 t+729\right)\right)\right)$ Copy content Toggle raw display
The polynomial $f_{1}$ is generic for any base field $K$ of characteristic $\neq$ 2,3