Properties

Label 46T26
46T26 1 8 1->8 27 1->27 2 7 2->7 28 2->28 3 19 3->19 36 3->36 4 20 4->20 35 4->35 5 31 5->31 43 5->43 6 32 6->32 44 6->44 7->5 7->43 8->6 8->44 9 14 9->14 10 13 10->13 11 22 11->22 11->22 12 21 12->21 12->21 29 13->29 33 13->33 30 14->30 34 14->34 15 38 15->38 45 15->45 16 37 16->37 46 16->46 17 17->12 17->46 18 18->11 18->45 19->7 23 19->23 20->8 24 20->24 21->16 21->35 22->15 22->36 23->1 24->2 25 25->13 25->32 26 26->14 26->31 27->25 40 27->40 28->26 39 28->39 29->2 29->38 30->1 30->37 31->3 31->9 32->4 32->10 33->16 33->17 34->15 34->18 35->25 35->27 36->26 36->28 37->34 37->39 38->33 38->40 39->6 42 39->42 40->5 41 40->41 41->4 41->18 42->3 42->17 43->12 43->30 44->11 44->29 45->19 45->42 46->20 46->41
Degree $46$
Order $1036288$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2^{12}:C_{23}:C_{11}$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(46, 26);
 
Copy content sage:G = TransitiveGroup(46, 26)
 
Copy content oscar:G = transitive_group(46, 26)
 
Copy content gap:G := TransitiveGroup(46, 26);
 

Group invariants

Abstract group:  $C_2^{12}:C_{23}:C_{11}$
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:  $1036288=2^{12} \cdot 11 \cdot 23$
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 $t$:  $26$
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(46).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(46), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(46), G));
 
Generators:  $(1,8,44,29,38,40,5,31,3,19,23)(2,7,43,30,37,39,6,32,4,20,24)(11,22,36,28,26,14,34,15,45,42,17,12,21,35,27,25,13,33,16,46,41,18)$, $(1,27,40,41,4,35,25,32,10,13,29,2,28,39,42,3,36,26,31,9,14,30)(5,43,12,21,16,37,34,18,45,19,7)(6,44,11,22,15,38,33,17,46,20,8)$
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)
$2$:  $C_2$
$11$:  $C_{11}$
$22$:  22T1
$253$:  $C_{23}:C_{11}$
$506$:  46T4
$518144$:  46T25

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 23: $C_{23}:C_{11}$

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

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