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Properties

Label 20T47
20T47 1 9 1->9 16 1->16 2 10 2->10 15 2->15 3 3->1 20 3->20 4 4->2 19 4->19 5 5->2 5->19 6 6->1 6->20 7 8 7->8 17 7->17 18 8->18 9->4 14 9->14 10->3 13 10->13 11 11->13 11->16 12 12->14 12->15 13->5 13->8 14->6 14->7 15->3 15->9 16->4 16->10 17->12 17->18 18->11 19->11 20->12
Degree $20$
Order $200$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_5^2:Q_8$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 47);
 
Copy content sage:G = TransitiveGroup(20, 47)
 
Copy content oscar:G = transitive_group(20, 47)
 
Copy content gap:G := TransitiveGroup(20, 47);
 

Group invariants

Abstract group:  $C_5^2:Q_8$
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:  $200=2^{3} \cdot 5^{2}$
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 $t$:  $47$
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);
 
$\card{\Aut(F/K)}$:  $2$
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:  $(1,9,14,6)(2,10,13,5)(3,20,12,15)(4,19,11,16)(7,8)(17,18)$, $(1,16,10,3)(2,15,9,4)(5,19)(6,20)(7,17,12,14)(8,18,11,13)$
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
$4$:  $C_2^2$
$8$:  $Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 10: $C_5^2 : Q_8$

Low degree siblings

10T20 x 3, 20T47 x 2, 25T17, 40T166 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 $2^{8},1^{4}$ $25$ $2$ $8$ $( 3,16)( 4,15)( 5,17)( 6,18)( 7,12)( 8,11)( 9,13)(10,14)$
4A $4^{4},2^{2}$ $50$ $4$ $14$ $( 1, 2)( 3,11,16, 8)( 4,12,15, 7)( 5, 9,17,13)( 6,10,18,14)(19,20)$
4B $4^{4},2^{2}$ $50$ $4$ $14$ $( 1,19, 5,16)( 2,20, 6,15)( 3,10,12,17)( 4, 9,11,18)( 7,14)( 8,13)$
4C $4^{4},2^{2}$ $50$ $4$ $14$ $( 1, 8, 5,20)( 2, 7, 6,19)( 3,13)( 4,14)( 9,12,18,16)(10,11,17,15)$
5A $5^{2},1^{10}$ $8$ $5$ $8$ $( 1, 5,10,14,17)( 2, 6, 9,13,18)$
5B $5^{4}$ $8$ $5$ $16$ $( 1,14, 5,17,10)( 2,13, 6,18, 9)( 3,16, 7,19,12)( 4,15, 8,20,11)$
5C $5^{4}$ $8$ $5$ $16$ $( 1,17,14,10, 5)( 2,18,13, 9, 6)( 3,16, 7,19,12)( 4,15, 8,20,11)$

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 4A 4B 4C 5A 5B 5C
Size 1 25 50 50 50 8 8 8
2 P 1A 1A 2A 2A 2A 5A 5B 5C
5 P 1A 2A 4A 4B 4C 1A 1A 1A
Type
200.44.1a R 1 1 1 1 1 1 1 1
200.44.1b R 1 1 −1 −1 1 1 1 1
200.44.1c R 1 1 −1 1 −1 1 1 1
200.44.1d R 1 1 1 −1 −1 1 1 1
200.44.2a S 2 −2 0 0 0 2 2 2
200.44.8a R 8 0 0 0 0 −2 −2 3
200.44.8b R 8 0 0 0 0 −2 3 −2
200.44.8c R 8 0 0 0 0 3 −2 −2

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