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Properties

Label 5T1
5T1 1 2 1->2 3 2->3 4 3->4 5 4->5 5->1
Degree $5$
Order $5$
Cyclic yes
Abelian yes
Solvable yes
Transitivity $1$
Primitive yes
$p$-group yes
Group: $C_5$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(5, 1);
 
Copy content sage:G = TransitiveGroup(5, 1)
 
Copy content oscar:G = transitive_group(5, 1)
 
Copy content gap:G := TransitiveGroup(5, 1);
 

Group invariants

Abstract group:  $C_5$
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:  $5$ (is prime)
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:  yes
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 $n$:  $5$
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 $t$:  $1$
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:   $C(5) = 5$
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:  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);
 
$\card{\Aut(F/K)}$:  $5$
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:  $(1,2,3,4,5)$
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

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 $1^{5}$ $1$ $1$ $0$ $()$
5A1 $5$ $1$ $5$ $4$ $(1,2,3,4,5)$
5A-1 $5$ $1$ $5$ $4$ $(1,5,4,3,2)$
5A2 $5$ $1$ $5$ $4$ $(1,3,5,2,4)$
5A-2 $5$ $1$ $5$ $4$ $(1,4,2,5,3)$

Malle's constant $a(G)$:     $1/4$

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 5A1 5A-1 5A2 5A-2
Size 1 1 1 1 1
5 P 1A 5A2 5A-2 5A-1 5A1
Type
5.1.1a R 1 1 1 1 1
5.1.1b1 C 1 ζ5−2 ζ52 ζ5 ζ5−1
5.1.1b2 C 1 ζ52 ζ5−2 ζ5−1 ζ5
5.1.1b3 C 1 ζ5−1 ζ5 ζ5−2 ζ52
5.1.1b4 C 1 ζ5 ζ5−1 ζ52 ζ5−2

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

Complete list of indecomposable integral representations:

Name Dim $(1,2,3,4,5) \mapsto $
Triv $1$ $\left(\begin{array}{r}1\end{array}\right)$
$J$ $4$ $\left(\begin{array}{rrrr}0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\end{array}\right)$
$R$ $5$ $\left(\begin{array}{rrrrr}0 & 1 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 1\\1 & 0 & 0 & 0 & 0\end{array}\right)$
The decomposition of an arbitrary integral representation as a direct sum of indecomposables is unique.

Regular extensions

$f_{ 1 } =$ $x^{5} - t^{2} x^{4} + \left(-20 t^{2} - 50\right) x^{3} + \left(4 t^{4} + 10 t^{2}\right) x^{2} + \left(24 t^{4} + 300 t^{2} + 625\right) x + \left(36 t^{4} + 95 t^{2}\right)$ Copy content Toggle raw display