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Properties

Label 20T5
20T5 1 4 1->4 6 1->6 2 3 2->3 5 2->5 8 3->8 18 3->18 7 4->7 17 4->17 9 5->9 12 5->12 10 6->10 11 6->11 7->5 7->11 8->6 8->12 13 9->13 19 9->19 14 10->14 20 10->20 11->13 16 11->16 12->14 15 12->15 13->8 13->17 14->7 14->18 15->1 15->19 16->2 16->20 17->2 17->15 18->1 18->16 19->3 19->10 20->4 20->9
Degree $20$
Order $20$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $F_5$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 5);
 
Copy content sage:G = TransitiveGroup(20, 5)
 
Copy content oscar:G = transitive_group(20, 5)
 
Copy content gap:G := TransitiveGroup(20, 5);
 

Group invariants

Abstract group:  $F_5$
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:  $20=2^{2} \cdot 5$
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:  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 $n$:  $20$
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 $t$:  $5$
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);
 
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);
 
$\card{\Aut(F/K)}$:  $20$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(20).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(20), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(20), G));
 
Generators:  $(1,4,17,15)(2,3,18,16)(5,12,14,7)(6,11,13,8)(9,19,10,20)$, $(1,6,10,14,18)(2,5,9,13,17)(3,8,12,15,19)(4,7,11,16,20)$
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)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 10: $F_5$

Low degree siblings

5T3, 10T4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{20}$ $1$ $1$ $0$ $()$
2A $2^{10}$ $5$ $2$ $10$ $( 1, 9)( 2,10)( 3, 7)( 4, 8)( 5, 6)(11,19)(12,20)(13,18)(14,17)(15,16)$
4A1 $4^{5}$ $5$ $4$ $15$ $( 1, 7, 9, 3)( 2, 8,10, 4)( 5,15, 6,16)(11,17,19,14)(12,18,20,13)$
4A-1 $4^{5}$ $5$ $4$ $15$ $( 1, 3, 9, 7)( 2, 4,10, 8)( 5,16, 6,15)(11,14,19,17)(12,13,20,18)$
5A $5^{4}$ $4$ $5$ $16$ $( 1, 6,10,14,18)( 2, 5, 9,13,17)( 3, 8,12,15,19)( 4, 7,11,16,20)$

Malle's constant $a(G)$:     $1/10$

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 4A1 4A-1 5A
Size 1 5 5 5 4
2 P 1A 1A 2A 2A 5A
5 P 1A 2A 4A1 4A-1 1A
Type
20.3.1a R 1 1 1 1 1
20.3.1b R 1 1 −1 −1 1
20.3.1c1 C 1 −1 −i i 1
20.3.1c2 C 1 −1 i −i 1
20.3.4a R 4 0 0 0 −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);
 

Regular extensions

Data not computed