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Properties

Label 20T15
20T15 1 6 1->6 17 1->17 2 5 2->5 18 2->18 3 4 3->4 8 3->8 7 4->7 5->7 9 5->9 6->8 10 6->10 11 7->11 12 8->12 14 9->14 20 9->20 13 10->13 19 10->19 11->14 16 11->16 12->13 15 12->15 13->17 14->18 15->16 15->20 16->19 17->1 18->2 19->4 20->3
Degree $20$
Order $60$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $A_5$

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, 15);
 
Copy content sage:G = TransitiveGroup(20, 15)
 
Copy content oscar:G = transitive_group(20, 15)
 
Copy content gap:G := TransitiveGroup(20, 15);
 

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 $n$:  $20$
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 $t$:  $15$
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);
 
$\card{\Aut(F/K)}$:  $2$
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:  $(1,17)(2,18)(3,4)(5,7)(6,8)(9,20)(10,19)(11,14)(12,13)(15,16)$, $(1,6,10,13,17)(2,5,9,14,18)(3,8,12,15,20)(4,7,11,16,19)$
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

Degree 2: None

Degree 4: None

Degree 5: $A_5$

Degree 10: $A_{5}$

Low degree siblings

5T4, 6T12, 10T7, 12T33, 15T5, 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 $1^{20}$ $1$ $1$ $0$ $()$
2A $2^{10}$ $15$ $2$ $10$ $( 1,14)( 2,13)( 3,11)( 4,12)( 5,10)( 6, 9)( 7, 8)(15,19)(16,20)(17,18)$
3A $3^{6},1^{2}$ $20$ $3$ $12$ $( 1, 8,13)( 2, 7,14)( 3, 6,19)( 4, 5,20)( 9,11,15)(10,12,16)$
5A1 $5^{4}$ $12$ $5$ $16$ $( 1,19,17,12, 8)( 2,20,18,11, 7)( 3,15, 5,10,14)( 4,16, 6, 9,13)$
5A2 $5^{4}$ $12$ $5$ $16$ $( 1,17, 8,19,12)( 2,18, 7,20,11)( 3, 5,14,15,10)( 4, 6,13,16, 9)$

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 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

Data not computed