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Properties

Label 20T24
20T24 1 14 1->14 20 1->20 2 13 2->13 19 2->19 3 10 3->10 3->19 4 9 4->9 4->20 5 5->3 18 5->18 6 6->4 17 6->17 7 7->4 7->13 8 8->3 8->14 9->1 9->8 10->2 10->7 11 11->7 11->18 12 12->8 12->17 13->6 13->12 14->5 14->11 15 15->1 15->11 16 16->2 16->12 17->9 17->15 18->10 18->16 19->6 19->15 20->5 20->16
Degree $20$
Order $100$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_5\times D_{10}$

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, 24);
 
Copy content sage:G = TransitiveGroup(20, 24)
 
Copy content oscar:G = transitive_group(20, 24)
 
Copy content gap:G := TransitiveGroup(20, 24);
 

Group invariants

Abstract group:  $C_5\times D_{10}$
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 $t$:  $24$
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:  $(1,20,5,3,10,7,13,12,17,15)(2,19,6,4,9,8,14,11,18,16)$, $(1,14,5,18,10,2,13,6,17,9)(3,19,15,11,7,4,20,16,12,8)$
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
$4$:  $C_2^2$
$5$:  $C_5$
$10$:  $D_{5}$, $C_{10}$ x 3
$20$:  $D_{10}$, 20T3
$50$:  $D_5\times C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 10: $D_5\times C_5$

Low degree siblings

20T24

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 $2^{10}$ $5$ $2$ $10$ $( 1,15)( 2,16)( 3,10)( 4, 9)( 5,20)( 6,19)( 7,13)( 8,14)(11,18)(12,17)$
2C $2^{10}$ $5$ $2$ $10$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,18)( 8,17)( 9,20)(10,19)$
5A1 $5^{4}$ $1$ $5$ $16$ $( 1,10,17, 5,13)( 2, 9,18, 6,14)( 3,12,20, 7,15)( 4,11,19, 8,16)$
5A-1 $5^{4}$ $1$ $5$ $16$ $( 1,13, 5,17,10)( 2,14, 6,18, 9)( 3,15, 7,20,12)( 4,16, 8,19,11)$
5A2 $5^{4}$ $1$ $5$ $16$ $( 1,17,13,10, 5)( 2,18,14, 9, 6)( 3,20,15,12, 7)( 4,19,16,11, 8)$
5A-2 $5^{4}$ $1$ $5$ $16$ $( 1, 5,10,13,17)( 2, 6, 9,14,18)( 3, 7,12,15,20)( 4, 8,11,16,19)$
5B1 $5^{4}$ $2$ $5$ $16$ $( 1,10,17, 5,13)( 2, 9,18, 6,14)( 3,15, 7,20,12)( 4,16, 8,19,11)$
5B2 $5^{4}$ $2$ $5$ $16$ $( 1,17,13,10, 5)( 2,18,14, 9, 6)( 3, 7,12,15,20)( 4, 8,11,16,19)$
5C1 $5^{2},1^{10}$ $2$ $5$ $8$ $( 1,17,13,10, 5)( 2,18,14, 9, 6)$
5C-1 $5^{2},1^{10}$ $2$ $5$ $8$ $( 1, 5,10,13,17)( 2, 6, 9,14,18)$
5C2 $5^{2},1^{10}$ $2$ $5$ $8$ $( 1,13, 5,17,10)( 2,14, 6,18, 9)$
5C-2 $5^{2},1^{10}$ $2$ $5$ $8$ $( 1,10,17, 5,13)( 2, 9,18, 6,14)$
5D1 $5^{4}$ $2$ $5$ $16$ $( 1, 5,10,13,17)( 2, 6, 9,14,18)( 3,12,20, 7,15)( 4,11,19, 8,16)$
5D-1 $5^{4}$ $2$ $5$ $16$ $( 1,17,13,10, 5)( 2,18,14, 9, 6)( 3,15, 7,20,12)( 4,16, 8,19,11)$
5D2 $5^{4}$ $2$ $5$ $16$ $( 1,10,17, 5,13)( 2, 9,18, 6,14)( 3,20,15,12, 7)( 4,19,16,11, 8)$
5D-2 $5^{4}$ $2$ $5$ $16$ $( 1,13, 5,17,10)( 2,14, 6,18, 9)( 3, 7,12,15,20)( 4, 8,11,16,19)$
10A1 $10^{2}$ $1$ $10$ $18$ $( 1, 6,10,14,17, 2, 5, 9,13,18)( 3, 8,12,16,20, 4, 7,11,15,19)$
10A-1 $10^{2}$ $1$ $10$ $18$ $( 1,18,13, 9, 5, 2,17,14,10, 6)( 3,19,15,11, 7, 4,20,16,12, 8)$
10A3 $10^{2}$ $1$ $10$ $18$ $( 1,14, 5,18,10, 2,13, 6,17, 9)( 3,16, 7,19,12, 4,15, 8,20,11)$
10A-3 $10^{2}$ $1$ $10$ $18$ $( 1, 9,17, 6,13, 2,10,18, 5,14)( 3,11,20, 8,15, 4,12,19, 7,16)$
10B1 $10^{2}$ $2$ $10$ $18$ $( 1, 6,10,14,17, 2, 5, 9,13,18)( 3,19,15,11, 7, 4,20,16,12, 8)$
10B3 $10^{2}$ $2$ $10$ $18$ $( 1,14, 5,18,10, 2,13, 6,17, 9)( 3,11,20, 8,15, 4,12,19, 7,16)$
10C1 $10,2^{5}$ $2$ $10$ $14$ $( 1, 9,17, 6,13, 2,10,18, 5,14)( 3, 4)( 7, 8)(11,12)(15,16)(19,20)$
10C-1 $10,2^{5}$ $2$ $10$ $14$ $( 1, 2)( 3,16, 7,19,12, 4,15, 8,20,11)( 5, 6)( 9,10)(13,14)(17,18)$
10C3 $10,2^{5}$ $2$ $10$ $14$ $( 1, 6,10,14,17, 2, 5, 9,13,18)( 3, 4)( 7, 8)(11,12)(15,16)(19,20)$
10C-3 $10,2^{5}$ $2$ $10$ $14$ $( 1, 2)( 3,19,15,11, 7, 4,20,16,12, 8)( 5, 6)( 9,10)(13,14)(17,18)$
10D1 $10^{2}$ $2$ $10$ $18$ $( 1,14, 5,18,10, 2,13, 6,17, 9)( 3, 8,12,16,20, 4, 7,11,15,19)$
10D-1 $10^{2}$ $2$ $10$ $18$ $( 1,18,13, 9, 5, 2,17,14,10, 6)( 3,11,20, 8,15, 4,12,19, 7,16)$
10D3 $10^{2}$ $2$ $10$ $18$ $( 1,18,13, 9, 5, 2,17,14,10, 6)( 3,16, 7,19,12, 4,15, 8,20,11)$
10D-3 $10^{2}$ $2$ $10$ $18$ $( 1, 9,17, 6,13, 2,10,18, 5,14)( 3, 8,12,16,20, 4, 7,11,15,19)$
10E1 $10^{2}$ $5$ $10$ $18$ $( 1, 7, 5,12,10,15,13,20,17, 3)( 2, 8, 6,11, 9,16,14,19,18, 4)$
10E-1 $10^{2}$ $5$ $10$ $18$ $( 1, 3,17,20,13,15,10,12, 5, 7)( 2, 4,18,19,14,16, 9,11, 6, 8)$
10E3 $10^{2}$ $5$ $10$ $18$ $( 1,12,13, 3, 5,15,17, 7,10,20)( 2,11,14, 4, 6,16,18, 8, 9,19)$
10E-3 $10^{2}$ $5$ $10$ $18$ $( 1,20,10, 7,17,15, 5, 3,13,12)( 2,19, 9, 8,18,16, 6, 4,14,11)$
10F1 $10^{2}$ $5$ $10$ $18$ $( 1, 4, 5, 8,10,11,13,16,17,19)( 2, 3, 6, 7, 9,12,14,15,18,20)$
10F-1 $10^{2}$ $5$ $10$ $18$ $( 1,19,17,16,13,11,10, 8, 5, 4)( 2,20,18,15,14,12, 9, 7, 6, 3)$
10F3 $10^{2}$ $5$ $10$ $18$ $( 1, 8,13,19, 5,11,17, 4,10,16)( 2, 7,14,20, 6,12,18, 3, 9,15)$
10F-3 $10^{2}$ $5$ $10$ $18$ $( 1,16,10, 4,17,11, 5,19,13, 8)( 2,15, 9, 3,18,12, 6,20,14, 7)$

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

40 x 40 character table

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