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Properties

Label 20T3
20T3 1 13 1->13 19 1->19 2 14 2->14 20 2->20 3 3->1 15 3->15 4 4->2 16 4->16 5 5->3 17 5->17 6 6->4 18 6->18 7 7->6 7->19 8 8->5 8->20 9 9->2 9->8 10 10->1 10->7 11 11->4 11->9 12 12->3 12->10 13->5 13->12 14->6 14->11 15->8 15->13 16->7 16->14 17->9 17->15 18->10 18->16 19->12 19->18 20->11 20->17
Degree $20$
Order $20$
Cyclic no
Abelian yes
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2\times C_{10}$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 3);
 
Copy content sage:G = TransitiveGroup(20, 3)
 
Copy content oscar:G = transitive_group(20, 3)
 
Copy content gap:G := TransitiveGroup(20, 3);
 

Group invariants

Abstract group:  $C_2\times C_{10}$
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:  $20=2^{2} \cdot 5$
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:  yes
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:  $1$
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 $t$:  $3$
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);
 
$\card{\Aut(F/K)}$:  $20$
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:  $(1,13,5,17,9,2,14,6,18,10)(3,15,8,20,11,4,16,7,19,12)$, $(1,19,18,16,14,11,9,8,5,3)(2,20,17,15,13,12,10,7,6,4)$
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
$4$:  $C_2^2$
$5$:  $C_5$
$10$:  $C_{10}$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $C_5$

Degree 10: $C_{10}$ x 3

Low degree siblings

There are no siblings 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 $2^{10}$ $1$ $2$ $10$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,19)(10,20)$
2C $2^{10}$ $1$ $2$ $10$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,18)( 8,17)( 9,20)(10,19)$
5A1 $5^{4}$ $1$ $5$ $16$ $( 1, 5, 9,14,18)( 2, 6,10,13,17)( 3, 8,11,16,19)( 4, 7,12,15,20)$
5A-1 $5^{4}$ $1$ $5$ $16$ $( 1,18,14, 9, 5)( 2,17,13,10, 6)( 3,19,16,11, 8)( 4,20,15,12, 7)$
5A2 $5^{4}$ $1$ $5$ $16$ $( 1, 9,18, 5,14)( 2,10,17, 6,13)( 3,11,19, 8,16)( 4,12,20, 7,15)$
5A-2 $5^{4}$ $1$ $5$ $16$ $( 1,14, 5,18, 9)( 2,13, 6,17,10)( 3,16, 8,19,11)( 4,15, 7,20,12)$
10A1 $10^{2}$ $1$ $10$ $18$ $( 1,13, 5,17, 9, 2,14, 6,18,10)( 3,15, 8,20,11, 4,16, 7,19,12)$
10A-1 $10^{2}$ $1$ $10$ $18$ $( 1,10,18, 6,14, 2, 9,17, 5,13)( 3,12,19, 7,16, 4,11,20, 8,15)$
10A3 $10^{2}$ $1$ $10$ $18$ $( 1,17,14,10, 5, 2,18,13, 9, 6)( 3,20,16,12, 8, 4,19,15,11, 7)$
10A-3 $10^{2}$ $1$ $10$ $18$ $( 1, 6, 9,13,18, 2, 5,10,14,17)( 3, 7,11,15,19, 4, 8,12,16,20)$
10B1 $10^{2}$ $1$ $10$ $18$ $( 1,16, 9, 3,18,11, 5,19,14, 8)( 2,15,10, 4,17,12, 6,20,13, 7)$
10B-1 $10^{2}$ $1$ $10$ $18$ $( 1, 8,14,19, 5,11,18, 3, 9,16)( 2, 7,13,20, 6,12,17, 4,10,15)$
10B3 $10^{2}$ $1$ $10$ $18$ $( 1, 3, 5, 8, 9,11,14,16,18,19)( 2, 4, 6, 7,10,12,13,15,17,20)$
10B-3 $10^{2}$ $1$ $10$ $18$ $( 1,19,18,16,14,11, 9, 8, 5, 3)( 2,20,17,15,13,12,10, 7, 6, 4)$
10C1 $10^{2}$ $1$ $10$ $18$ $( 1, 4, 5, 7, 9,12,14,15,18,20)( 2, 3, 6, 8,10,11,13,16,17,19)$
10C-1 $10^{2}$ $1$ $10$ $18$ $( 1,20,18,15,14,12, 9, 7, 5, 4)( 2,19,17,16,13,11,10, 8, 6, 3)$
10C3 $10^{2}$ $1$ $10$ $18$ $( 1, 7,14,20, 5,12,18, 4, 9,15)( 2, 8,13,19, 6,11,17, 3,10,16)$
10C-3 $10^{2}$ $1$ $10$ $18$ $( 1,15, 9, 4,18,12, 5,20,14, 7)( 2,16,10, 3,17,11, 6,19,13, 8)$

Malle's constant $a(G)$:     $1/10$

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 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 10B1 10B-1 10B3 10B-3 10C1 10C-1 10C3 10C-3
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 1A 1A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 5A2 5A-2 5A1 5A-1 5A1 5A-1 5A-2 5A2
5 P 1A 2A 2B 2C 1A 1A 1A 1A 2A 2A 2A 2A 2B 2B 2B 2B 2C 2C 2C 2C
Type
20.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
20.5.1b R 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
20.5.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
20.5.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
20.5.1e1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2
20.5.1e2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52
20.5.1e3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1
20.5.1e4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5
20.5.1f1 C 1 −1 −1 1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ5−1 ζ5 ζ52 ζ5−2
20.5.1f2 C 1 −1 −1 1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52
20.5.1f3 C 1 −1 −1 1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1
20.5.1f4 C 1 −1 −1 1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5−2 ζ52 ζ5−1 ζ5
20.5.1g1 C 1 −1 1 −1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2
20.5.1g2 C 1 −1 1 −1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52
20.5.1g3 C 1 −1 1 −1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1
20.5.1g4 C 1 −1 1 −1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5
20.5.1h1 C 1 1 −1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2
20.5.1h2 C 1 1 −1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52
20.5.1h3 C 1 1 −1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1
20.5.1h4 C 1 1 −1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5

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