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Properties

Label 38T21
38T21 1 13 1->13 20 1->20 2 14 2->14 37 2->37 3 15 3->15 35 3->35 4 16 4->16 33 4->33 5 17 5->17 31 5->31 6 18 6->18 29 6->29 7 19 7->19 27 7->27 8 8->1 25 8->25 9 9->2 23 9->23 10 10->3 21 10->21 11 11->4 38 11->38 12 12->5 36 12->36 13->6 34 13->34 14->7 32 14->32 15->8 30 15->30 16->9 28 16->28 17->10 26 17->26 18->11 24 18->24 19->12 22 19->22 20->19 20->36 21->6 21->24 22->12 22->31 23->18 23->38 24->5 24->26 25->11 25->33 26->21 27->4 27->28 28->10 28->35 29->16 29->23 30->3 31->9 31->37 32->15 32->25 33->2 33->32 34->8 34->20 35->14 35->27 36->1 36->34 37->7 37->22 38->13 38->29
Degree $38$
Order $6498$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{19}^2:(C_3\times S_3)$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(38, 21);
 
Copy content sage:G = TransitiveGroup(38, 21)
 
Copy content oscar:G = transitive_group(38, 21)
 
Copy content gap:G := TransitiveGroup(38, 21);
 

Group invariants

Abstract group:  $C_{19}^2:(C_3\times S_3)$
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:  $6498=2 \cdot 3^{2} \cdot 19^{2}$
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:  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 $t$:  $21$
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(38).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(38), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(38), G));
 
Generators:  $(1,13,6,18,11,4,16,9,2,14,7,19,12,5,17,10,3,15,8)(20,36,34)(21,24,26)(22,31,37)(23,38,29)(25,33,32)(27,28,35)$, $(1,20,19,22,12,36)(2,37,7,27,4,33)(3,35,14,32,15,30)(5,31,9,23,18,24)(6,29,16,28,10,21)(8,25,11,38,13,34)(17,26)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: None

Low degree siblings

There are no siblings 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

54 x 54 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

Data not computed