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Properties

Label 20T41
20T41 1 2 1->2 6 1->6 5 2->5 3 7 3->7 4 8 4->8 9 5->9 10 6->10 11 7->11 12 8->12 14 9->14 19 9->19 13 10->13 20 10->20 11->12 16 11->16 15 12->15 17 13->17 18 14->18 15->19 16->20 17->1 18->2 19->4 20->3
Degree $20$
Order $160$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2\wr C_5$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 41);
 
Copy content sage:G = TransitiveGroup(20, 41)
 
Copy content oscar:G = transitive_group(20, 41)
 
Copy content gap:G := TransitiveGroup(20, 41);
 

Group invariants

Abstract group:  $C_2\wr C_5$
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:  $160=2^{5} \cdot 5$
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 $t$:  $41$
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);
 
$\card{\Aut(F/K)}$:  $4$
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:  $(1,6,10,13,17)(2,5,9,14,18)(3,7,11,16,20)(4,8,12,15,19)$, $(1,2)(9,19)(10,20)(11,12)$
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)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$
$80$:  $C_2^4 : C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_2^4 : C_5$, $C_2 \times (C_2^4 : C_5)$

Low degree siblings

10T14 x 3, 20T40 x 12, 20T41 x 5, 20T44 x 3, 20T46 x 3, 32T2133, 40T121 x 6, 40T122 x 6, 40T123 x 12, 40T141, 40T142 x 3

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

Malle's constant $a(G)$:     $1/4$

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 2D 2E 2F 2G 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3
Size 1 1 5 5 5 5 5 5 16 16 16 16 16 16 16 16
2 P 1A 1A 1A 1A 1A 1A 1A 1A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2
5 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 1A 1A 2A 2A 2A 2A
Type
160.235.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
160.235.1b R 1 −1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
160.235.1c1 C 1 1 1 1 1 1 1 1 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−1 ζ5−2 ζ5
160.235.1c2 C 1 1 1 1 1 1 1 1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ5 ζ52 ζ5−1
160.235.1c3 C 1 1 1 1 1 1 1 1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ52 ζ5−1 ζ5−2
160.235.1c4 C 1 1 1 1 1 1 1 1 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5−2 ζ5 ζ52
160.235.1d1 C 1 −1 −1 1 −1 −1 1 1 ζ5−2 ζ52 ζ5−1 ζ5 −ζ52 −ζ5−1 −ζ5−2 −ζ5
160.235.1d2 C 1 −1 −1 1 −1 −1 1 1 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5−2 −ζ5 −ζ52 −ζ5−1
160.235.1d3 C 1 −1 −1 1 −1 −1 1 1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5 −ζ52 −ζ5−1 −ζ5−2
160.235.1d4 C 1 −1 −1 1 −1 −1 1 1 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5−1 −ζ5−2 −ζ5 −ζ52
160.235.5a R 5 5 −3 1 1 1 1 −3 0 0 0 0 0 0 0 0
160.235.5b R 5 5 1 −3 −3 1 1 1 0 0 0 0 0 0 0 0
160.235.5c R 5 5 1 1 1 −3 −3 1 0 0 0 0 0 0 0 0
160.235.5d R 5 −5 −1 −3 3 −1 1 1 0 0 0 0 0 0 0 0
160.235.5e R 5 −5 −1 1 −1 3 −3 1 0 0 0 0 0 0 0 0
160.235.5f R 5 −5 3 1 −1 −1 1 −3 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