Properties

Label 46T15
46T15 1 30 1->30 31 1->31 2 41 2->41 42 2->42 3 28 3->28 3->31 4 38 4->38 43 4->43 5 25 5->25 32 5->32 6 35 6->35 44 6->44 7 33 7->33 45 7->45 8 8->32 8->45 9 34 9->34 9->42 10 29 10->29 46 10->46 11 11->35 39 11->39 12 24 12->24 26 12->26 13 36 13->36 13->36 14 14->25 14->46 15 15->33 37 15->37 16 16->26 16->43 17 17->30 17->38 18 27 18->27 40 18->40 19 19->27 19->39 20 20->28 20->37 21 21->24 21->40 22 22->29 22->34 23 23->41 23->44 24->3 24->3 25->8 25->9 26->13 26->15 27->18 27->21 28->4 28->23 29->5 29->10 30->10 30->16 31->15 31->22 32->20 33->2 33->11 34->7 34->17 35->12 35->23 36->6 36->17 37->12 37->22 38->18 39->1 39->9 40->7 40->14 41->13 41->19 42->1 42->19 43->2 43->6 44->8 44->11 45->14 45->16 46->20 46->21
Degree $46$
Order $23276$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_{23}:F_{23}$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(46, 15);
 
Copy content sage:G = TransitiveGroup(46, 15)
 
Copy content oscar:G = transitive_group(46, 15)
 
Copy content gap:G := TransitiveGroup(46, 15);
 

Group invariants

Abstract group:  $D_{23}:F_{23}$
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 $t$:  $15$
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:  $(1,30,16,26,15,37,12,24,3,31,22,29,10,46,20,28,4,43,2,42,19,39)(5,32)(6,44,8,45,14,25,9,34,17,38,18,27,21,40,7,33,11,35,23,41,13,36)$, $(1,31,15,33,2,41,19,27,18,40,14,46,21,24,3,28,23,44,11,39,9,42)(4,38)(5,25,8,32,20,37,22,34,7,45,16,43,6,35,12,26,13,36,17,30,10,29)$
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
$4$:  $C_2^2$
$11$:  $C_{11}$
$22$:  22T1 x 3
$44$:  44T2
$506$:  $F_{23}$ x 2
$1012$:  46T6 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 23: None

Low degree siblings

46T15 x 10

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

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