Properties

Label 10T4
10T4 1 2 1->2 3 1->3 4 2->4 9 2->9 5 3->5 6 3->6 4->3 4->6 7 5->7 10 5->10 6->7 8 6->8 7->4 7->9 8->1 8->10 9->1 9->8 10->2
Degree $10$
Order $20$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $F_5$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(10, 4);
 
Copy content sage:G = TransitiveGroup(10, 4)
 
Copy content oscar:G = transitive_group(10, 4)
 
Copy content gap:G := TransitiveGroup(10, 4);
 

Group invariants

Abstract group:  $F_5$
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:  $20=2^{2} \cdot 5$
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:  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 $n$:  $10$
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 $t$:  $4$
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);
 
CHM label:   $1/2[F(5)]2$
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(10).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(10), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(10), G));
 
Generators:  $(1,3,5,7,9)(2,4,6,8,10)$, $(1,2,9,8)(3,6,7,4)(5,10)$
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)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $F_5$

Low degree siblings

5T3, 20T5

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{10}$ $1$ $1$ $0$ $()$
2A $2^{4},1^{2}$ $5$ $2$ $4$ $( 1, 5)( 2, 4)( 6,10)( 7, 9)$
4A1 $4^{2},2$ $5$ $4$ $7$ $( 1, 4, 5, 2)( 3, 8)( 6, 9,10, 7)$
4A-1 $4^{2},2$ $5$ $4$ $7$ $( 1, 2, 5, 4)( 3, 8)( 6, 7,10, 9)$
5A $5^{2}$ $4$ $5$ $8$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$

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 4A1 4A-1 5A
Size 1 5 5 5 4
2 P 1A 1A 2A 2A 5A
5 P 1A 2A 4A1 4A-1 1A
Type
20.3.1a R 1 1 1 1 1
20.3.1b R 1 1 −1 −1 1
20.3.1c1 C 1 −1 −i i 1
20.3.1c2 C 1 −1 i −i 1
20.3.4a R 4 0 0 0 −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

$f_{ 1 } =$ $x^{10} - 5 x^{9} + \left(-t^{2} + t + 17\right) x^{8} + \left(4 t^{2} - 4 t - 38\right) x^{7} + \left(-8 t^{2} + 3 t + 66\right) x^{6} + \left(10 t^{2} + 5 t - 86\right) x^{5} + \left(-9 t^{2} - 9 t + 69\right) x^{4} + \left(6 t^{2} + 5 t - 29\right) x^{3} + \left(-2 t^{2} - 3 t + 4\right) x^{2} + \left(2 t + 1\right) x - 1$ Copy content Toggle raw display