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Properties

Label 20T36
20T36 1 20 1->20 1->20 2 19 2->19 2->19 3 6 3->6 16 3->16 4 5 4->5 15 4->15 8 5->8 10 5->10 7 6->7 9 6->9 7->1 7->3 8->2 8->4 14 9->14 17 9->17 13 10->13 18 10->18 11 11->8 11->10 12 12->7 12->9 13->1 13->3 14->2 14->4 15->5 16->6 17->12 17->12 18->11 18->11 19->14 19->17 20->13 20->18
Degree $20$
Order $120$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2\times A_5$

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, 36);
 
Copy content sage:G = TransitiveGroup(20, 36)
 
Copy content oscar:G = transitive_group(20, 36)
 
Copy content gap:G := TransitiveGroup(20, 36);
 

Group invariants

Abstract group:  $C_2\times A_5$
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:  $120=2^{3} \cdot 3 \cdot 5$
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 $t$:  $36$
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:  $(1,20,13)(2,19,14)(3,6,7)(4,5,8)(9,17,12)(10,18,11)$, $(1,20,18,11,8,2,19,17,12,7)(3,16,6,9,14,4,15,5,10,13)$
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)
$2$:  $C_2$
$60$:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 10: $A_{5}$

Low degree siblings

10T11, 12T75, 12T76, 20T31, 24T203, 30T29, 30T30, 40T61

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 $2^{10}$ $1$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2B $2^{8},1^{4}$ $15$ $2$ $8$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7, 9)( 8,10)(13,18)(14,17)$
2C $2^{10}$ $15$ $2$ $10$ $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,19)( 8,20)( 9,15)(10,16)(11,18)(12,17)$
3A $3^{6},1^{2}$ $20$ $3$ $12$ $( 1,10, 4)( 2, 9, 3)( 5,14,12)( 6,13,11)(15,18,19)(16,17,20)$
5A1 $5^{4}$ $12$ $5$ $16$ $( 1,14,11,17, 3)( 2,13,12,18, 4)( 5,10,19, 8,15)( 6, 9,20, 7,16)$
5A2 $5^{4}$ $12$ $5$ $16$ $( 1,11, 3,14,17)( 2,12, 4,13,18)( 5,19,15,10, 8)( 6,20,16, 9, 7)$
6A $6^{3},2$ $20$ $6$ $16$ $( 1, 3,10, 2, 4, 9)( 5,11,14, 6,12,13)( 7, 8)(15,20,18,16,19,17)$
10A1 $10^{2}$ $12$ $10$ $18$ $( 1,18,14, 4,11, 2,17,13, 3,12)( 5, 7,10,16,19, 6, 8, 9,15,20)$
10A3 $10^{2}$ $12$ $10$ $18$ $( 1, 4,17,12,14, 2, 3,18,11,13)( 5,16, 8,20,10, 6,15, 7,19, 9)$

Malle's constant $a(G)$:     $1/8$

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 2B 2C 3A 5A1 5A2 6A 10A1 10A3
Size 1 1 15 15 20 12 12 20 12 12
2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 5A1 5A2
3 P 1A 2A 2B 2C 1A 5A2 5A1 2A 10A3 10A1
5 P 1A 2A 2B 2C 3A 1A 1A 6A 2A 2A
Type
120.35.1a R 1 1 1 1 1 1 1 1 1 1
120.35.1b R 1 −1 −1 1 1 1 1 −1 −1 −1
120.35.3a1 R 3 3 −1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5
120.35.3a2 R 3 3 −1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52
120.35.3b1 R 3 −3 1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5
120.35.3b2 R 3 −3 1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52
120.35.4a R 4 4 0 0 1 −1 −1 1 −1 −1
120.35.4b R 4 −4 0 0 1 −1 −1 −1 1 1
120.35.5a R 5 5 1 1 −1 0 0 −1 0 0
120.35.5b R 5 −5 −1 1 −1 0 0 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