Properties

Label 35T37
35T37 1 18 1->18 28 1->28 2 17 2->17 23 2->23 3 19 3->19 25 3->25 4 21 4->21 27 4->27 5 16 5->16 22 5->22 6 20 6->20 24 6->24 7 15 7->15 26 7->26 8 8->28 33 8->33 9 9->27 34 9->34 10 10->24 29 10->29 11 11->26 32 11->32 12 12->25 31 12->31 13 13->23 35 13->35 14 14->22 30 14->30 15->7 15->30 16->3 16->35 17->1 17->29 18->2 18->31 19->5 19->32 20->4 20->34 21->6 21->33 22->4 22->14 23->6 23->11 24->2 24->12 25->3 25->10 26->1 26->13 27->5 27->8 28->7 28->9 29->10 29->17 30->8 30->21 31->11 31->19 32->12 32->18 33->14 33->15 34->13 34->16 35->9 35->20
Degree $35$
Order $24010$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_7^4:C_{10}$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(35, 37);
 
Copy content sage:G = TransitiveGroup(35, 37)
 
Copy content oscar:G = transitive_group(35, 37)
 
Copy content gap:G := TransitiveGroup(35, 37);
 

Group invariants

Abstract group:  $C_7^4:C_{10}$
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:  $24010=2 \cdot 5 \cdot 7^{4}$
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 $n$:  $35$
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 $t$:  $37$
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);
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(35).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(35), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(35), G));
 
Generators:  $(1,18,31,11,26)(2,17,29,10,24)(3,19,32,12,25)(4,21,33,14,22)(5,16,35,9,27)(6,20,34,13,23)(7,15,30,8,28)$, $(1,28,9,34,16,3,25,10,29,17)(2,23,11,32,18)(4,27,8,33,15,7,26,13,35,20)(5,22,14,30,21,6,24,12,31,19)$
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)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$

Degree 7: None

Low degree siblings

35T37 x 79

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Conjugacy classes not computed

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

Character table not computed

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