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Properties

Label 20T45
20T45 1 5 1->5 19 1->19 2 6 2->6 20 2->20 3 4 3->4 17 3->17 18 4->18 16 5->16 15 6->15 7 7->4 9 7->9 8 8->3 10 8->10 12 9->12 11 10->11 11->9 11->15 12->10 12->16 13 13->7 14 13->14 14->8 17->14 17->19 18->13 18->20 19->2 20->1
Degree $20$
Order $160$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2^4:D_5$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 45);
 
Copy content sage:G = TransitiveGroup(20, 45)
 
Copy content oscar:G = transitive_group(20, 45)
 
Copy content gap:G := TransitiveGroup(20, 45);
 

Group invariants

Abstract group:  $C_2^4:D_5$
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:  $160=2^{5} \cdot 5$
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 $t$:  $45$
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);
 
$\card{\Aut(F/K)}$:  $4$
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:  $(1,5)(2,6)(3,4)(7,9)(8,10)(11,15)(12,16)(13,14)(17,19)(18,20)$, $(1,19,2,20)(3,17,14,8)(4,18,13,7)(5,16)(6,15)(9,12,10,11)$
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)
$2$:  $C_2$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $D_{5}$

Degree 10: $(C_2^4 : C_5) : C_2$, $(C_2^4 : C_5) : C_2$ x 2

Low degree siblings

10T15 x 3, 10T16 x 3, 16T415, 20T38 x 6, 20T39, 20T43 x 3, 20T45 x 2, 32T2132, 40T143 x 3, 40T144 x 3, 40T145 x 6, 40T146

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 $2^{8},1^{4}$ $5$ $2$ $8$ $( 1, 2)( 3,14)( 4,13)( 7,18)( 8,17)( 9,10)(11,12)(19,20)$
2B $2^{8},1^{4}$ $5$ $2$ $8$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)$
2C $2^{8},1^{4}$ $5$ $2$ $8$ $( 3, 4)( 5,15)( 6,16)( 7,17)( 8,18)( 9,10)(13,14)(19,20)$
2D $2^{10}$ $20$ $2$ $10$ $( 1,14)( 2,13)( 3,12)( 4,11)( 5,19)( 6,20)( 7, 8)( 9,15)(10,16)(17,18)$
4A $4^{4},2^{2}$ $20$ $4$ $14$ $( 1,19, 2,20)( 3,17,14, 8)( 4,18,13, 7)( 5,16)( 6,15)( 9,12,10,11)$
4B $4^{4},1^{4}$ $20$ $4$ $12$ $( 1, 8,11,18)( 2, 7,12,17)( 3,15,14, 6)( 4,16,13, 5)$
4C $4^{4},2^{2}$ $20$ $4$ $14$ $( 1,11)( 2,12)( 3,20, 4,19)( 5,18,15, 8)( 6,17,16, 7)( 9,13,10,14)$
5A1 $5^{4}$ $32$ $5$ $16$ $( 1,15,19,14,17)( 2,16,20,13,18)( 3, 8,12, 6,10)( 4, 7,11, 5, 9)$
5A2 $5^{4}$ $32$ $5$ $16$ $( 1,19,17,15,14)( 2,20,18,16,13)( 3,12,10, 8, 6)( 4,11, 9, 7, 5)$

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 2D 4A 4B 4C 5A1 5A2
Size 1 5 5 5 20 20 20 20 32 32
2 P 1A 1A 1A 1A 1A 2A 2B 2C 5A2 5A1
5 P 1A 2A 2B 2C 2D 4A 4B 4C 1A 1A
Type
160.234.1a R 1 1 1 1 1 1 1 1 1 1
160.234.1b R 1 1 1 1 −1 −1 −1 −1 1 1
160.234.2a1 R 2 2 2 2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5
160.234.2a2 R 2 2 2 2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52
160.234.5a R 5 −3 1 1 1 1 −1 −1 0 0
160.234.5b R 5 1 −3 1 1 −1 1 −1 0 0
160.234.5c R 5 1 1 −3 1 −1 −1 1 0 0
160.234.5d R 5 −3 1 1 −1 −1 1 1 0 0
160.234.5e R 5 1 −3 1 −1 1 −1 1 0 0
160.234.5f R 5 1 1 −3 −1 1 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