Properties

Label 20T30
20T30 1 3 1->3 6 1->6 2 4 2->4 5 2->5 8 3->8 7 4->7 9 5->9 13 5->13 10 6->10 14 6->14 11 7->11 12 8->12 9->14 10->13 16 11->16 15 12->15 17 13->17 18 14->18 15->17 20 15->20 16->18 19 16->19 17->1 18->2 19->4 19->20 20->3
Degree $20$
Order $120$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $S_5$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 30);
 
Copy content sage:G = TransitiveGroup(20, 30)
 
Copy content oscar:G = transitive_group(20, 30)
 
Copy content gap:G := TransitiveGroup(20, 30);
 

Group invariants

Abstract group:  $S_5$
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 $t$:  $30$
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:  $(1,3)(2,4)(5,13)(6,14)(15,17)(16,18)(19,20)$, $(1,6,10,13,17)(2,5,9,14,18)(3,8,12,15,20)(4,7,11,16,19)$
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

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $S_5$

Degree 10: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62

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 $2^{7},1^{6}$ $10$ $2$ $7$ $( 1,14)( 2,13)( 3, 6)( 4, 5)( 7, 8)( 9,11)(10,12)$
2B $2^{10}$ $15$ $2$ $10$ $( 1,18)( 2,17)( 3,16)( 4,15)( 5,13)( 6,14)( 7,12)( 8,11)( 9,10)(19,20)$
3A $3^{6},1^{2}$ $20$ $3$ $12$ $( 1, 9, 3)( 2,10, 4)( 5,13,12)( 6,14,11)(15,18,20)(16,17,19)$
4A $4^{5}$ $30$ $4$ $15$ $( 1,15,18, 4)( 2,16,17, 3)( 5, 7,13,12)( 6, 8,14,11)( 9,19,10,20)$
5A $5^{4}$ $24$ $5$ $16$ $( 1, 6,15, 4,12)( 2, 5,16, 3,11)( 7,14,19,17,10)( 8,13,20,18, 9)$
6A $6^{2},3^{2},2$ $20$ $6$ $15$ $( 1, 6, 9,14, 3,11)( 2, 5,10,13, 4,12)( 7, 8)(15,20,18)(16,19,17)$

Malle's constant $a(G)$:     $1/7$

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 3A 4A 5A 6A
Size 1 10 15 20 30 24 20
2 P 1A 1A 1A 3A 2B 5A 3A
3 P 1A 2A 2B 1A 4A 5A 2A
5 P 1A 2A 2B 3A 4A 1A 6A
Type
120.34.1a R 1 1 1 1 1 1 1
120.34.1b R 1 −1 1 1 −1 1 −1
120.34.4a R 4 2 0 1 0 −1 −1
120.34.4b R 4 −2 0 1 0 −1 1
120.34.5a R 5 −1 1 −1 1 0 −1
120.34.5b R 5 1 1 −1 −1 0 1
120.34.6a R 6 0 −2 0 0 1 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