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Properties

Label 10T43
10T43 1 6 1->6 2 4 2->4 7 2->7 10 2->10 3 8 3->8 4->6 9 4->9 5 5->10 6->8 8->10 10->2
Degree $10$
Order $28800$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $S_5^2 \wr C_2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(10, 43);
 
Copy content sage:G = TransitiveGroup(10, 43)
 
Copy content oscar:G = transitive_group(10, 43)
 
Copy content gap:G := TransitiveGroup(10, 43);
 

Group invariants

Abstract group:  $S_5^2 \wr C_2$
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:  $28800=2^{7} \cdot 3^{2} \cdot 5^{2}$
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:  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 $t$:  $43$
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:   $[S(5)^{2}]2$
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);
 
$\card{\Aut(F/K)}$:  $1$
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:  $(2,4,6,8,10)$, $(1,6)(2,7)(3,8)(4,9)(5,10)$, $(2,10)$
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)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

12T288, 20T539, 20T540, 20T542, 20T544, 24T13996, 24T13997, 24T13998, 25T106, 30T1011, 36T13308, 40T14374, 40T14375, 40T14376, 40T14377, 40T14378, 40T14379, 40T14380, 40T14381

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 $2,1^{8}$ $20$ $2$ $1$ $( 4,10)$
2B $2^{2},1^{6}$ $30$ $2$ $2$ $( 4, 8)( 6,10)$
2C $2^{2},1^{6}$ $100$ $2$ $2$ $(1,9)(2,8)$
2D $2^{5}$ $120$ $2$ $5$ $( 1, 6)( 2, 9)( 3, 4)( 5, 8)( 7,10)$
2E $2^{4},1^{2}$ $225$ $2$ $4$ $( 1, 5)( 2,10)( 3, 9)( 4, 6)$
2F $2^{3},1^{4}$ $300$ $2$ $3$ $( 3, 9)( 4,10)( 5, 7)$
3A $3,1^{7}$ $40$ $3$ $2$ $(4,6,8)$
3B $3^{2},1^{4}$ $400$ $3$ $4$ $( 3, 7, 5)( 4, 6,10)$
4A $4,1^{6}$ $60$ $4$ $3$ $( 4, 6, 8,10)$
4B $4,2,1^{4}$ $600$ $4$ $4$ $(1,5,3,9)(2,6)$
4C $4^{2},1^{2}$ $900$ $4$ $6$ $( 1, 9, 5, 3)( 2, 4,10, 6)$
4D $4,2^{2},1^{2}$ $900$ $4$ $5$ $(2,6,4,8)(3,5)(7,9)$
4E $4,2^{3}$ $1200$ $4$ $6$ $( 1, 2, 9, 8)( 3, 4)( 5,10)( 6, 7)$
4F $4^{2},2$ $1800$ $4$ $7$ $( 1, 4, 5, 8)( 2, 7)( 3,10, 9, 6)$
5A $5,1^{5}$ $48$ $5$ $4$ $(1,9,7,5,3)$
5B $5^{2}$ $576$ $5$ $8$ $( 1, 7, 5, 9, 3)( 2, 4, 6,10, 8)$
6A $3,2,1^{5}$ $40$ $6$ $3$ $( 2,10, 6)( 4, 8)$
6B $3^{2},2^{2}$ $400$ $6$ $6$ $( 1, 9)( 2, 8)( 3, 5, 7)( 4,10, 6)$
6C $3,2^{2},1^{3}$ $400$ $6$ $4$ $( 2,10)( 4, 8, 6)( 5, 9)$
6D $3,2,1^{5}$ $400$ $6$ $3$ $( 1, 7)( 4,10, 6)$
6E $3,2^{2},1^{3}$ $600$ $6$ $4$ $(1,3)(4,8,6)(5,7)$
6F $3,2^{3},1$ $600$ $6$ $5$ $( 2, 8, 6)( 3, 9)( 4,10)( 5, 7)$
6G $3^{2},2,1^{2}$ $800$ $6$ $5$ $( 2, 6, 8)( 3, 7, 9)( 4,10)$
6H $6,2^{2}$ $2400$ $6$ $7$ $( 1, 4, 3, 8, 5, 2)( 6, 7)( 9,10)$
8A $8,2$ $3600$ $8$ $8$ $( 1, 4, 9,10, 5, 6, 3, 2)( 7, 8)$
10A $5,2,1^{3}$ $480$ $10$ $5$ $(1,3,7,9,5)(4,8)$
10B $5,2^{2},1$ $720$ $10$ $6$ $( 1, 5, 9, 3, 7)( 4, 8)( 6,10)$
10C $10$ $2880$ $10$ $9$ $( 1, 2, 7, 4, 5, 6, 9,10, 3, 8)$
12A $4,3,1^{3}$ $1200$ $12$ $5$ $(1,7,3,5)(4,6,8)$
12B $4,3,2,1$ $1200$ $12$ $6$ $( 1, 9, 3, 5)( 2, 6)( 4, 8,10)$
12C $6,4$ $2400$ $12$ $8$ $( 1, 8, 9, 2)( 3, 6, 5, 4, 7,10)$
15A $5,3,1^{2}$ $960$ $15$ $6$ $( 1, 5, 9, 7, 3)( 2,10, 6)$
20A $5,4,1$ $1440$ $20$ $7$ $( 1, 3, 5, 7, 9)( 4, 6, 8,10)$
30A $5,3,2$ $960$ $30$ $7$ $( 1, 7, 5, 3, 9)( 2, 6,10)( 4, 8)$

Malle's constant $a(G)$:     $1$

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

35 x 35 character table

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

$f_{ 1 } =$ $x^{10} + 2 x^{5} + t x^{2} + 1$ Copy content Toggle raw display