Properties

Label 46T22
46T22 1 36 1->36 45 1->45 2 29 2->29 2->45 3 31 3->31 3->36 4 40 4->40 43 4->43 5 26 5->26 27 5->27 6 34 6->34 35 6->35 7 41 7->41 44 7->44 8 25 8->25 30 8->30 9 32 9->32 39 9->39 10 10->25 10->39 11 11->34 46 11->46 12 12->30 12->43 13 13->29 37 13->37 14 38 14->38 14->44 15 24 15->24 28 15->28 16 33 16->33 16->35 17 42 17->42 17->42 18 18->26 18->28 19 19->33 19->37 20 20->40 20->46 21 21->24 21->32 22 22->31 22->41 23 23->27 23->38 24->11 24->17 25->3 25->18 26->18 26->19 27->10 27->20 28->2 28->21 29->17 29->22 30->9 30->23 31->1 31->1 32->2 32->16 33->3 33->8 34->4 34->23 35->5 35->15 36->6 36->7 37->7 37->22 38->8 39->6 39->9 40->10 40->21 41->11 41->13 42->5 42->12 43->13 43->20 44->12 45->4 45->15 46->16 46->19
Degree $46$
Order $256036$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{23}^2:C_{11}:C_{44}$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(46, 22);
 
Copy content sage:G = TransitiveGroup(46, 22)
 
Copy content oscar:G = transitive_group(46, 22)
 
Copy content gap:G := TransitiveGroup(46, 22);
 

Group invariants

Abstract group:  $C_{23}^2:C_{11}:C_{44}$
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:  $256036=2^{2} \cdot 11^{2} \cdot 23^{2}$
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 $t$:  $22$
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:  $(1,45,15,28,21,24,17,42,12,30,23,38,8,25,18,26,19,33,3,36,6,34,4,43,13,37,7,41,11,46,16,35,5,27,20,40,10,39,9,32,2,29,22,31)(14,44)$, $(1,36,7,44,12,43,20,46,19,37,22,41,13,29,17,42,5,26,18,28,2,45,4,40,21,32,16,33,8,30,9,39,6,35,15,24,11,34,23,27,10,25,3,31)(14,38)$
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)
$2$:  $C_2$
$4$:  $C_4$
$11$:  $C_{11}$
$22$:  $D_{11}$, 22T1
$44$:  $C_{44}$, 44T3
$242$:  22T7
$484$:  44T26

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

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