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Properties

Label 10T34
10T34 1 3 1->3 2 4 2->4 2->4 7 2->7 5 3->5 6 4->6 10 4->10 5->7 5->7 5->10 8 6->8 9 7->9 7->9 8->10 9->1 9->5 10->2 10->2
Degree $10$
Order $960$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2^4 : A_5$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(10, 34);
 
Copy content sage:G = TransitiveGroup(10, 34)
 
Copy content oscar:G = transitive_group(10, 34)
 
Copy content gap:G := TransitiveGroup(10, 34);
 

Group invariants

Abstract group:  $C_2^4 : A_5$
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:  $960=2^{6} \cdot 3 \cdot 5$
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:  no
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 $t$:  $34$
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:   $[2^{4}]A(5)$
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:  $(2,4,10)(5,7,9)$, $(1,3,5,7,9)(2,4,6,8,10)$, $(2,7)(5,10)$
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)
$60$:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $A_5$

Low degree siblings

16T1081, 20T172, 20T177, 30T214, 30T217, 40T888, 40T889, 40T932, 40T942, 40T944, 40T945

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 $2^{4},1^{2}$ $5$ $2$ $4$ $( 1, 6)( 2, 7)( 3, 8)( 5,10)$
2B $2^{2},1^{6}$ $10$ $2$ $2$ $( 4, 9)( 5,10)$
2C $2^{4},1^{2}$ $60$ $2$ $4$ $( 1, 8)( 2, 5)( 3, 6)( 7,10)$
3A $3^{2},1^{4}$ $80$ $3$ $4$ $( 2, 3, 5)( 7, 8,10)$
4A $4^{2},1^{2}$ $60$ $4$ $6$ $( 1, 9, 6, 4)( 3, 5, 8,10)$
4B $4,2^{3}$ $120$ $4$ $6$ $( 1, 2)( 3, 8)( 4,10, 9, 5)( 6, 7)$
5A1 $5^{2}$ $192$ $5$ $8$ $( 1, 8, 5, 7, 4)( 2, 9, 6, 3,10)$
5A2 $5^{2}$ $192$ $5$ $8$ $( 1, 5, 4, 8, 7)( 2, 6,10, 9, 3)$
6A $3^{2},2^{2}$ $80$ $6$ $6$ $( 1, 5, 3)( 2, 7)( 4, 9)( 6,10, 8)$
6B1 $6,2,1^{2}$ $80$ $6$ $6$ $( 1, 6)( 2,10, 3, 7, 5, 8)$
6B-1 $6,2,1^{2}$ $80$ $6$ $6$ $( 1, 6)( 2, 8, 5, 7, 3,10)$

Malle's constant $a(G)$:     $1/2$

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 3A 4A 4B 5A1 5A2 6A 6B1 6B-1
Size 1 5 10 60 80 60 120 192 192 80 80 80
2 P 1A 1A 1A 1A 3A 2A 2B 5A2 5A1 3A 3A 3A
3 P 1A 2A 2B 2C 1A 4A 4B 5A2 5A1 2B 2A 2A
5 P 1A 2A 2B 2C 3A 4A 4B 1A 1A 6A 6B-1 6B1
Type
960.11358.1a R 1 1 1 1 1 1 1 1 1 1 1 1
960.11358.3a1 R 3 3 3 −1 0 −1 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0
960.11358.3a2 R 3 3 3 −1 0 −1 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0
960.11358.4a R 4 4 4 0 1 0 0 −1 −1 1 1 1
960.11358.5a R 5 5 5 1 −1 1 1 0 0 −1 −1 −1
960.11358.5b R 5 −3 1 1 2 1 −1 0 0 −2 0 0
960.11358.5c1 C 5 −3 1 1 −1 1 −1 0 0 1 −1−2ζ3 1+2ζ3
960.11358.5c2 C 5 −3 1 1 −1 1 −1 0 0 1 1+2ζ3 −1−2ζ3
960.11358.10a R 10 2 −2 2 1 −2 0 0 0 1 −1 −1
960.11358.10b R 10 2 −2 −2 1 2 0 0 0 1 −1 −1
960.11358.15a R 15 −9 3 −1 0 −1 1 0 0 0 0 0
960.11358.20a R 20 4 −4 0 −1 0 0 0 0 −1 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

$f_{ 1 } =$ $x^{10} + 75 x^{6} - 3 t^{2} x^{4} - 9 t^{2}$ Copy content Toggle raw display