Properties

Label 38T48
38T48 1 15 1->15 38 1->38 2 16 2->16 37 2->37 3 3->1 18 3->18 4 4->2 17 4->17 5 5->3 19 5->19 6 6->4 20 6->20 7 7->5 21 7->21 8 8->6 22 8->22 9 9->7 23 9->23 10 10->8 24 10->24 11 11->10 26 11->26 12 12->9 25 12->25 13 13->12 28 13->28 14 14->11 27 14->27 15->13 29 15->29 16->14 30 16->30 17->16 32 17->32 18->15 31 18->31 19->17 33 19->33 20->18 34 20->34 21->19 35 21->35 22->20 36 22->36 23->21 23->38 24->22 24->37 25->2 25->24 26->1 26->23 27->3 27->25 28->4 28->26 29->6 29->27 30->5 30->28 31->8 31->29 32->7 32->30 33->10 33->31 34->9 34->32 35->11 35->34 36->12 36->33 37->14 37->35 38->13 38->36
Degree $38$
Order $4980736$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2^{18}.C_{19}$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(38, 48);
 
Copy content sage:G = TransitiveGroup(38, 48)
 
Copy content oscar:G = transitive_group(38, 48)
 
Copy content gap:G := TransitiveGroup(38, 48);
 

Group invariants

Abstract group:  $C_2^{18}.C_{19}$
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:  $4980736=2^{18} \cdot 19$
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 $n$:  $38$
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 $t$:  $48$
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(38).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(38), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(38), G));
 
Generators:  $(1,15,29,6,20,34,9,23,38,13,28,4,17,32,7,21,35,11,26)(2,16,30,5,19,33,10,24,37,14,27,3,18,31,8,22,36,12,25)$, $(1,38,36,33,31,29,27,25,24,22,20,18,15,13,12,9,7,5,3)(2,37,35,34,32,30,28,26,23,21,19,17,16,14,11,10,8,6,4)$
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)
$19$:  $C_{19}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 19: $C_{19}$

Low degree siblings

38T48 x 13796

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