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Properties

Label 5T4
5T4 1 2 1->2 3 2->3 3->1 4 3->4 5 4->5 5->3
Degree $5$
Order $60$
Cyclic no
Abelian no
Solvable no
Transitivity $3$
Primitive yes
$p$-group no
Group: $A_5$

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This is the smallest instance of a non-solvable Galois group, implying that the general quintic cannot be solvable in radicals and motivating the creation of Galois theory.

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(5, 4);
 
Copy content sage:G = TransitiveGroup(5, 4)
 
Copy content oscar:G = transitive_group(5, 4)
 
Copy content gap:G := TransitiveGroup(5, 4);
 

Group invariants

Abstract group:  $A_5$
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:  $60=2^{2} \cdot 3 \cdot 5$
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:  no
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 $n$:  $5$
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:   $A5$
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:  3
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(5).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(5), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(5), G));
 
Generators:  $(1,2,3)$, $(3,4,5)$
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

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

6T12, 10T7, 12T33, 15T5, 20T15, 30T9

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{5}$ $1$ $1$ $0$ $()$
2A $2^{2},1$ $15$ $2$ $2$ $(1,4)(2,5)$
3A $3,1^{2}$ $20$ $3$ $2$ $(2,3,5)$
5A1 $5$ $12$ $5$ $4$ $(1,4,2,5,3)$
5A2 $5$ $12$ $5$ $4$ $(1,2,3,4,5)$

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 3A 5A1 5A2
Size 1 15 20 12 12
2 P 1A 1A 3A 5A2 5A1
3 P 1A 2A 1A 5A2 5A1
5 P 1A 2A 3A 1A 1A
Type
60.5.1a R 1 1 1 1 1
60.5.3a1 R 3 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52
60.5.3a2 R 3 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5
60.5.4a R 4 0 1 −1 −1
60.5.5a R 5 1 −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

$f_{ 1 } =$ $x^{5} + 75 x^{3} + t x^{2} + 3 t$ Copy content Toggle raw display
$f_{ 2 } =$ $x^{5} + 125 \left(5 s^{2}-t^{2}+36\right)^{2}/\left(4 \left(t \left(5 s^{2}-t^{2}+36\right)^{2}-52 t \left(5 s^{2}-t^{2}+36\right)+576 t-10 \left(5 s^{2}-t^{2}+36\right)^{2}+360 \left(5 s^{2}-t^{2}+36\right)-3456\right)\right) x^{3} + 3125 \left(5 s^{2}-t^{2}+36\right)^{5}/\left(256 \left(t \left(5 s^{2}-t^{2}+36\right)^{2}-52 t \left(5 s^{2}-t^{2}+36\right)+576 t-10 \left(5 s^{2}-t^{2}+36\right)^{2}+360 \left(5 s^{2}-t^{2}+36\right)-3456\right)^{2}\right) x + 3125 \left(5 s^{2}-t^{2}+36\right)^{5}/\left(256 \left(t \left(5 s^{2}-t^{2}+36\right)^{2}-52 t \left(5 s^{2}-t^{2}+36\right)+576 t-10 \left(5 s^{2}-t^{2}+36\right)^{2}+360 \left(5 s^{2}-t^{2}+36\right)-3456\right)^{2}\right)$ Copy content Toggle raw display
The polynomial $f_{2}$ is generic for the base field $\Q$