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Properties

Label 20T14
20T14 1 8 1->8 12 1->12 2 5 2->5 11 2->11 3 7 3->7 10 3->10 4 6 4->6 9 4->9 5->12 16 5->16 6->9 15 6->15 7->11 14 7->14 8->10 13 8->13 9->16 20 9->20 10->13 19 10->19 11->15 18 11->18 12->14 17 12->17 13->4 13->20 14->3 14->17 15->2 15->19 16->1 16->18 17->4 17->8 18->1 18->7 19->3 19->6 20->2 20->5
Degree $20$
Order $60$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_5\times A_4$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 14);
 
Copy content sage:G = TransitiveGroup(20, 14)
 
Copy content oscar:G = transitive_group(20, 14)
 
Copy content gap:G := TransitiveGroup(20, 14);
 

Group invariants

Abstract group:  $C_5\times A_4$
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:  $60=2^{2} \cdot 3 \cdot 5$
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 $t$:  $14$
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);
 
$\card{\Aut(F/K)}$:  $5$
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:  $(1,8,10,13,20,2,5,12,14,17,4,6,9,16,18)(3,7,11,15,19)$, $(1,12,17,8,13,4,9,20,5,16)(2,11,18,7,14,3,10,19,6,15)$
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)
$3$:  $C_3$
$5$:  $C_5$
$12$:  $A_4$
$15$:  $C_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 5: $C_5$

Degree 10: None

Low degree siblings

30T11

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 $2^{10}$ $3$ $2$ $10$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)$
3A1 $3^{5},1^{5}$ $4$ $3$ $10$ $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)$
3A-1 $3^{5},1^{5}$ $4$ $3$ $10$ $( 2, 4, 3)( 6, 8, 7)(10,12,11)(14,16,15)(18,20,19)$
5A1 $5^{4}$ $1$ $5$ $16$ $( 1,13, 5,17, 9)( 2,14, 6,18,10)( 3,15, 7,19,11)( 4,16, 8,20,12)$
5A-1 $5^{4}$ $1$ $5$ $16$ $( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$
5A2 $5^{4}$ $1$ $5$ $16$ $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 7,11,15,19)( 4, 8,12,16,20)$
5A-2 $5^{4}$ $1$ $5$ $16$ $( 1,17,13, 9, 5)( 2,18,14,10, 6)( 3,19,15,11, 7)( 4,20,16,12, 8)$
10A1 $10^{2}$ $3$ $10$ $18$ $( 1,20,13,12, 5, 4,17,16, 9, 8)( 2,19,14,11, 6, 3,18,15,10, 7)$
10A-1 $10^{2}$ $3$ $10$ $18$ $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3, 8,11,16,19, 4, 7,12,15,20)$
10A3 $10^{2}$ $3$ $10$ $18$ $( 1,12,17, 8,13, 4, 9,20, 5,16)( 2,11,18, 7,14, 3,10,19, 6,15)$
10A-3 $10^{2}$ $3$ $10$ $18$ $( 1,14, 5,18, 9, 2,13, 6,17,10)( 3,16, 7,20,11, 4,15, 8,19,12)$
15A1 $15,5$ $4$ $15$ $18$ $( 1,13, 5,17, 9)( 2,16, 7,18,12, 3,14, 8,19,10, 4,15, 6,20,11)$
15A-1 $15,5$ $4$ $15$ $18$ $( 1, 9,17, 5,13)( 2,11,20, 6,15, 4,10,19, 8,14, 3,12,18, 7,16)$
15A2 $15,5$ $4$ $15$ $18$ $( 1, 5, 9,13,17)( 2, 7,12,14,19, 4, 6,11,16,18, 3, 8,10,15,20)$
15A-2 $15,5$ $4$ $15$ $18$ $( 1,20,14, 9, 8, 2,17,16,10, 5, 4,18,13,12, 6)( 3,19,15,11, 7)$
15A4 $15,5$ $4$ $15$ $18$ $( 1, 9,17, 5,13)( 2,12,19, 6,16, 3,10,20, 7,14, 4,11,18, 8,15)$
15A-4 $15,5$ $4$ $15$ $18$ $( 1,13, 5,17, 9)( 2,15, 8,18,11, 4,14, 7,20,10, 3,16, 6,19,12)$
15A7 $15,5$ $4$ $15$ $18$ $( 1, 5, 9,13,17)( 2, 8,11,14,20, 3, 6,12,15,18, 4, 7,10,16,19)$
15A-7 $15,5$ $4$ $15$ $18$ $( 1,20,15, 9, 8, 3,17,16,11, 5, 4,19,13,12, 7)( 2,18,14,10, 6)$

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 3A1 3A-1 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 15A1 15A-1 15A2 15A-2 15A4 15A-4 15A7 15A-7
Size 1 3 4 4 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4
2 P 1A 1A 3A-1 3A1 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 15A2 15A-2 15A4 15A-4 15A-7 15A7 15A-1 15A1
3 P 1A 2A 1A 1A 5A-2 5A2 5A1 5A-1 10A3 10A-3 10A-1 10A1 5A-2 5A2 5A1 5A-1 5A2 5A-2 5A1 5A-1
5 P 1A 2A 3A-1 3A1 1A 1A 1A 1A 2A 2A 2A 2A 3A1 3A-1 3A-1 3A1 3A1 3A-1 3A1 3A-1
Type
60.9.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
60.9.1b1 C 1 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
60.9.1b2 C 1 1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
60.9.1c1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1
60.9.1c2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5
60.9.1c3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52
60.9.1c4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2
60.9.1d1 C 1 1 ζ15−5 ζ155 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−3 ζ153 ζ156 ζ15−6 ζ15−1 ζ15 ζ15−2 ζ152 ζ15−4 ζ154 ζ15−7 ζ157
60.9.1d2 C 1 1 ζ155 ζ15−5 ζ156 ζ15−6 ζ15−3 ζ153 ζ153 ζ15−3 ζ15−6 ζ156 ζ15 ζ15−1 ζ152 ζ15−2 ζ154 ζ15−4 ζ157 ζ15−7
60.9.1d3 C 1 1 ζ15−5 ζ155 ζ156 ζ15−6 ζ15−3 ζ153 ζ153 ζ15−3 ζ15−6 ζ156 ζ15−4 ζ154 ζ157 ζ15−7 ζ15−1 ζ15 ζ152 ζ15−2
60.9.1d4 C 1 1 ζ155 ζ15−5 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−3 ζ153 ζ156 ζ15−6 ζ154 ζ15−4 ζ15−7 ζ157 ζ15 ζ15−1 ζ15−2 ζ152
60.9.1d5 C 1 1 ζ15−5 ζ155 ζ15−3 ζ153 ζ15−6 ζ156 ζ156 ζ15−6 ζ153 ζ15−3 ζ152 ζ15−2 ζ154 ζ15−4 ζ15−7 ζ157 ζ15−1 ζ15
60.9.1d6 C 1 1 ζ155 ζ15−5 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−6 ζ156 ζ15−3 ζ153 ζ15−2 ζ152 ζ15−4 ζ154 ζ157 ζ15−7 ζ15 ζ15−1
60.9.1d7 C 1 1 ζ15−5 ζ155 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−6 ζ156 ζ15−3 ζ153 ζ15−7 ζ157 ζ15 ζ15−1 ζ152 ζ15−2 ζ15−4 ζ154
60.9.1d8 C 1 1 ζ155 ζ15−5 ζ15−3 ζ153 ζ15−6 ζ156 ζ156 ζ15−6 ζ153 ζ15−3 ζ157 ζ15−7 ζ15−1 ζ15 ζ15−2 ζ152 ζ154 ζ15−4
60.9.3a R 3 −1 0 0 3 3 3 3 −1 −1 −1 −1 0 0 0 0 0 0 0 0
60.9.3b1 C 3 −1 0 0 3ζ5−2 3ζ52 3ζ5 3ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 0 0 0 0 0 0 0 0
60.9.3b2 C 3 −1 0 0 3ζ52 3ζ5−2 3ζ5−1 3ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 0 0 0 0 0 0 0 0
60.9.3b3 C 3 −1 0 0 3ζ5−1 3ζ5 3ζ5−2 3ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 0 0 0 0 0 0 0 0
60.9.3b4 C 3 −1 0 0 3ζ5 3ζ5−1 3ζ52 3ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 0 0 0 0 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