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Properties

Label 20T9
20T9 1 15 1->15 17 1->17 2 16 2->16 18 2->18 3 3->1 20 3->20 4 4->2 19 4->19 5 5->1 8 5->8 6 6->2 7 6->7 7->3 14 7->14 8->4 13 8->13 9 9->6 9->19 10 10->5 10->20 11 11->5 11->8 12 12->6 12->7 13->9 13->11 14->10 14->12 15->12 15->17 16->11 16->18 17->3 17->13 18->4 18->14 19->10 19->15 20->9 20->16
Degree $20$
Order $40$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2\times F_5$

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, 9);
 
Copy content sage:G = TransitiveGroup(20, 9)
 
Copy content oscar:G = transitive_group(20, 9)
 
Copy content gap:G := TransitiveGroup(20, 9);
 

Group invariants

Abstract group:  $C_2\times F_5$
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:  $40=2^{3} \cdot 5$
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 $t$:  $9$
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:  $(1,17,13,9,6,2,18,14,10,5)(3,20,16,11,8,4,19,15,12,7)$, $(1,15,17,3)(2,16,18,4)(5,8,13,11)(6,7,14,12)(9,19,10,20)$
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
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 10: $F_5$

Low degree siblings

10T5 x 2, 20T13, 40T14

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 $2^{8},1^{4}$ $5$ $2$ $8$ $( 1,13)( 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)(15,20)(16,19)$
2C $2^{10}$ $5$ $2$ $10$ $( 1,17)( 2,18)( 3,15)( 4,16)( 5,13)( 6,14)( 7,12)( 8,11)( 9,10)(19,20)$
4A1 $4^{5}$ $5$ $4$ $15$ $( 1,15,17, 3)( 2,16,18, 4)( 5, 8,13,11)( 6, 7,14,12)( 9,19,10,20)$
4A-1 $4^{5}$ $5$ $4$ $15$ $( 1, 3,17,15)( 2, 4,18,16)( 5,11,13, 8)( 6,12,14, 7)( 9,20,10,19)$
4B1 $4^{5}$ $5$ $4$ $15$ $( 1,12, 2,11)( 3,17,20, 6)( 4,18,19, 5)( 7,10,16,14)( 8, 9,15,13)$
4B-1 $4^{5}$ $5$ $4$ $15$ $( 1,20, 5, 8)( 2,19, 6, 7)( 3,13, 4,14)( 9,16,18,11)(10,15,17,12)$
5A $5^{4}$ $4$ $5$ $16$ $( 1,13, 6,18,10)( 2,14, 5,17, 9)( 3,16, 8,19,12)( 4,15, 7,20,11)$
10A $10^{2}$ $4$ $10$ $18$ $( 1,17,13, 9, 6, 2,18,14,10, 5)( 3,20,16,11, 8, 4,19,15,12, 7)$

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

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 5A 10A
Size 1 1 5 5 5 5 5 5 4 4
2 P 1A 1A 1A 1A 2C 2C 2C 2C 5A 5A
5 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 1A 2A
Type
40.12.1a R 1 1 1 1 1 1 1 1 1 1
40.12.1b R 1 −1 −1 1 −1 −1 1 1 1 −1
40.12.1c R 1 −1 −1 1 1 1 −1 −1 1 −1
40.12.1d R 1 1 1 1 −1 −1 −1 −1 1 1
40.12.1e1 C 1 −1 1 −1 −i i i −i 1 −1
40.12.1e2 C 1 −1 1 −1 i −i −i i 1 −1
40.12.1f1 C 1 1 −1 −1 −i i −i i 1 1
40.12.1f2 C 1 1 −1 −1 i −i i −i 1 1
40.12.4a R 4 4 0 0 0 0 0 0 −1 −1
40.12.4b R 4 −4 0 0 0 0 0 0 −1 1

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