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Properties

Label 20T28
20T28 1 5 1->5 12 1->12 2 6 2->6 11 2->11 3 3->6 20 3->20 4 4->5 19 4->19 15 5->15 16 6->16 7 9 7->9 7->16 8 10 8->10 8->15 17 9->17 9->20 18 10->18 10->19 11->12 13 11->13 14 12->14 13->4 13->14 14->3 15->18 16->17 17->8 18->7 19->1 20->2
Degree $20$
Order $100$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_5^2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 28);
 
Copy content sage:G = TransitiveGroup(20, 28)
 
Copy content oscar:G = transitive_group(20, 28)
 
Copy content gap:G := TransitiveGroup(20, 28);
 

Group invariants

Abstract group:  $D_5^2$
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:  $100=2^{2} \cdot 5^{2}$
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 $t$:  $28$
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);
 
$\card{\Aut(F/K)}$:  $10$
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:  $(1,5)(2,6)(3,20)(4,19)(7,16)(8,15)(9,17)(10,18)(11,12)(13,14)$, $(1,12,14,3,6,16,17,8,10,19)(2,11,13,4,5,15,18,7,9,20)$
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)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$10$:  $D_{5}$ x 2
$20$:  $D_{10}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 10: $D_5^2$

Low degree siblings

10T9 x 2, 20T28, 25T12

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 $2^{10}$ $5$ $2$ $10$ $( 1,19)( 2,20)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)(15,18)(16,17)$
2B $2^{10}$ $5$ $2$ $10$ $( 1, 4)( 2, 3)( 5,19)( 6,20)( 7,17)( 8,18)( 9,16)(10,15)(11,14)(12,13)$
2C $2^{10}$ $25$ $2$ $10$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5, 6)( 7, 8)(13,17)(14,18)(15,19)(16,20)$
5A1 $5^{4}$ $2$ $5$ $16$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3, 8,12,16,19)( 4, 7,11,15,20)$
5A2 $5^{4}$ $2$ $5$ $16$ $( 1,10,17, 6,14)( 2, 9,18, 5,13)( 3,12,19, 8,16)( 4,11,20, 7,15)$
5B1 $5^{4}$ $2$ $5$ $16$ $( 1,14, 6,17,10)( 2,13, 5,18, 9)( 3,12,19, 8,16)( 4,11,20, 7,15)$
5B2 $5^{4}$ $2$ $5$ $16$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,19,16,12, 8)( 4,20,15,11, 7)$
5C1 $5^{2},1^{10}$ $4$ $5$ $8$ $( 3,12,19, 8,16)( 4,11,20, 7,15)$
5C2 $5^{2},1^{10}$ $4$ $5$ $8$ $( 3,19,16,12, 8)( 4,20,15,11, 7)$
5D1 $5^{4}$ $4$ $5$ $16$ $( 1,17,14,10, 6)( 2,18,13, 9, 5)( 3,16, 8,19,12)( 4,15, 7,20,11)$
5D2 $5^{4}$ $4$ $5$ $16$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,16, 8,19,12)( 4,15, 7,20,11)$
10A1 $10^{2}$ $10$ $10$ $18$ $( 1,12, 6,16,10,19,14, 3,17, 8)( 2,11, 5,15, 9,20,13, 4,18, 7)$
10A3 $10^{2}$ $10$ $10$ $18$ $( 1,16,14, 8, 6,19,17,12,10, 3)( 2,15,13, 7, 5,20,18,11, 9, 4)$
10B1 $10^{2}$ $10$ $10$ $18$ $( 1, 7,14,15, 6, 4,17,11,10,20)( 2, 8,13,16, 5, 3,18,12, 9,19)$
10B3 $10^{2}$ $10$ $10$ $18$ $( 1,11, 6, 7,10, 4,14,20,17,15)( 2,12, 5, 8, 9, 3,13,19,18,16)$

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 5A1 5A2 5B1 5B2 5C1 5C2 5D1 5D2 10A1 10A3 10B1 10B3
Size 1 5 5 25 2 2 2 2 4 4 4 4 10 10 10 10
2 P 1A 1A 1A 1A 5A2 5A1 5B2 5B1 5C2 5C1 5D2 5D1 5A1 5A2 5B1 5B2
5 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2B 2B
Type
100.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
100.13.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
100.13.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1
100.13.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 1 1
100.13.2a1 R 2 0 2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 0
100.13.2a2 R 2 0 2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 0
100.13.2b1 R 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52
100.13.2b2 R 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5
100.13.2c1 R 2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
100.13.2c2 R 2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
100.13.2d1 R 2 0 −2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
100.13.2d2 R 2 0 −2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0
100.13.4a1 R 4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 ζ5−2+2+ζ52 −1 −ζ5−2+1−ζ52 0 0 0 0
100.13.4a2 R 4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 −ζ5−2+1−ζ52 −1 ζ5−2+2+ζ52 0 0 0 0
100.13.4b1 R 4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −ζ5−2+1−ζ52 −1 ζ5−2+2+ζ52 −1 0 0 0 0
100.13.4b2 R 4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 ζ5−2+2+ζ52 −1 −ζ5−2+1−ζ52 −1 0 0 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