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Properties

Label 20T10
20T10 1 3 1->3 19 1->19 2 4 2->4 20 2->20 5 3->5 18 3->18 6 4->6 17 4->17 8 5->8 16 5->16 7 6->7 15 6->15 9 7->9 13 7->13 10 8->10 14 8->14 12 9->12 9->12 11 10->11 10->11 11->14 12->13 13->15 14->16 15->17 16->18 17->20 18->19 19->2 20->1
Degree $20$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{20}$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 10);
 
Copy content sage:G = TransitiveGroup(20, 10)
 
Copy content oscar:G = transitive_group(20, 10)
 

Group invariants

Abstract group:  $D_{20}$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $40=2^{3} \cdot 5$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar: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)
 
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)
 
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)
 
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
 

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)
 
Transitive number $t$:  $10$
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]
 
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)
 
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)
 
$\card{\Aut(F/K)}$:  $2$
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])
 
Generators:  $(1,19)(2,20)(3,18)(4,17)(5,16)(6,15)(7,13)(8,14)(9,12)(10,11)$, $(1,3,5,8,10,11,14,16,18,19,2,4,6,7,9,12,13,15,17,20)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$10$:  $D_{5}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 10: $D_{10}$

Low degree siblings

20T10, 40T12

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 $2^{10}$ $10$ $2$ $10$ $( 1, 3)( 2, 4)( 5,20)( 6,19)( 7,18)( 8,17)( 9,16)(10,15)(11,13)(12,14)$
2C $2^{9},1^{2}$ $10$ $2$ $9$ $( 1,18)( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)(19,20)$
4A $4^{5}$ $2$ $4$ $15$ $( 1,11, 2,12)( 3,14, 4,13)( 5,16, 6,15)( 7,17, 8,18)( 9,20,10,19)$
5A1 $5^{4}$ $2$ $5$ $16$ $( 1,10,18, 6,13)( 2, 9,17, 5,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$
5A2 $5^{4}$ $2$ $5$ $16$ $( 1,18,13,10, 6)( 2,17,14, 9, 5)( 3,19,15,11, 7)( 4,20,16,12, 8)$
10A1 $10^{2}$ $2$ $10$ $18$ $( 1, 5,10,14,18, 2, 6, 9,13,17)( 3, 8,11,16,19, 4, 7,12,15,20)$
10A3 $10^{2}$ $2$ $10$ $18$ $( 1,14, 6,17,10, 2,13, 5,18, 9)( 3,16, 7,20,11, 4,15, 8,19,12)$
20A1 $20$ $2$ $20$ $19$ $( 1, 3, 5, 8,10,11,14,16,18,19, 2, 4, 6, 7, 9,12,13,15,17,20)$
20A3 $20$ $2$ $20$ $19$ $( 1, 8,14,19, 6,12,17, 3,10,16, 2, 7,13,20, 5,11,18, 4, 9,15)$
20A7 $20$ $2$ $20$ $19$ $( 1, 7,14,20, 6,11,17, 4,10,15, 2, 8,13,19, 5,12,18, 3, 9,16)$
20A9 $20$ $2$ $20$ $19$ $( 1,19,17,16,13,11, 9, 8, 6, 3, 2,20,18,15,14,12,10, 7, 5, 4)$

Malle's constant $a(G)$:     $1/9$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 

Character table

1A 2A 2B 2C 4A 5A1 5A2 10A1 10A3 20A1 20A3 20A7 20A9
Size 1 1 10 10 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 5A2 5A1 5A1 5A2 10A1 10A3 10A3 10A1
5 P 1A 2A 2B 2C 4A 1A 1A 2A 2A 4A 4A 4A 4A
Type
40.6.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
40.6.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1
40.6.1c R 1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
40.6.1d R 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
40.6.2a R 2 −2 0 0 0 2 2 −2 −2 0 0 0 0
40.6.2b1 R 2 2 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
40.6.2b2 R 2 2 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
40.6.2c1 R 2 2 0 0 −2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
40.6.2c2 R 2 2 0 0 −2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
40.6.2d1 R 2 −2 0 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ20−4−ζ204 ζ20−2+ζ202 −ζ20−3−ζ203 ζ20−1+ζ20 −ζ20−1−ζ20 ζ20−3+ζ203
40.6.2d2 R 2 −2 0 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ20−4−ζ204 ζ20−2+ζ202 ζ20−3+ζ203 −ζ20−1−ζ20 ζ20−1+ζ20 −ζ20−3−ζ203
40.6.2d3 R 2 −2 0 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 ζ20−2+ζ202 −ζ20−4−ζ204 −ζ20−1−ζ20 −ζ20−3−ζ203 ζ20−3+ζ203 ζ20−1+ζ20
40.6.2d4 R 2 −2 0 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 ζ20−2+ζ202 −ζ20−4−ζ204 ζ20−1+ζ20 ζ20−3+ζ203 −ζ20−3−ζ203 −ζ20−1−ζ20

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 

Regular extensions

Data not computed