Properties

Label 46T13
46T13 1 23 1->23 39 1->39 2 11 2->11 30 2->30 3 22 3->22 44 3->44 4 10 4->10 35 4->35 5 21 5->21 26 5->26 6 9 6->9 40 6->40 7 20 7->20 31 7->31 8 45 8->45 19 9->19 36 9->36 10->7 27 10->27 18 11->18 41 11->41 12 12->6 32 12->32 13 17 13->17 46 13->46 14 14->5 37 14->37 15 16 15->16 28 15->28 16->4 42 16->42 17->15 33 17->33 18->3 24 18->24 19->14 38 19->38 20->2 29 20->29 21->13 43 21->43 22->1 34 22->34 23->12 25 23->25 24->22 24->40 25->4 25->38 26->9 26->36 27->14 27->34 28->19 28->32 29->1 29->30 30->6 30->28 31->11 31->26 32->16 32->24 33->21 33->45 34->3 34->43 35->8 35->41 36->13 36->39 37->18 38->23 38->35 39->5 39->33 40->10 40->31 41->15 41->29 42->20 42->27 43->2 43->25 44->7 44->46 45->12 45->44 46->17 46->42
Degree $46$
Order $23276$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{23}^2:D_{22}$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(46, 13);
 
Copy content sage:G = TransitiveGroup(46, 13)
 
Copy content oscar:G = transitive_group(46, 13)
 
Copy content gap:G := TransitiveGroup(46, 13);
 

Group invariants

Abstract group:  $C_{23}^2:D_{22}$
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:  $23276=2^{2} \cdot 11 \cdot 23^{2}$
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 $n$:  $46$
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 $t$:  $13$
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(46).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(46), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(46), G));
 
Generators:  $(1,39,5,26,9,36,13,46,17,33,21,43,2,30,6,40,10,27,14,37,18,24,22,34,3,44,7,31,11,41,15,28,19,38,23,25,4,35,8,45,12,32,16,42,20,29)$, $(1,23,12,6,9,19,14,5,21,13,17,15,16,4,10,7,20,2,11,18,3,22)(24,40,31,26,36,39,33,45,44,46,42,27,34,43,25,38,35,41,29,30,28,32)$
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
$4$:  $C_2^2$
$22$:  $D_{11}$
$44$:  $D_{22}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 23: 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

59 x 59 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