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Properties

Label 46T45
46T45 1 40 1->40 44 1->44 2 39 2->39 43 2->43 3 10 3->10 13 3->13 4 9 4->9 14 4->14 5 15 5->15 5->40 6 16 6->16 6->39 7 41 7->41 42 7->42 8 8->41 8->42 28 9->28 34 9->34 27 10->27 33 10->33 11 31 11->31 37 11->37 12 32 12->32 38 12->38 29 13->29 13->33 30 14->30 14->34 45 15->45 46 15->46 16->45 16->46 17 24 17->24 25 17->25 18 23 18->23 26 18->26 19 19->6 19->17 20 20->5 20->18 21 21->12 21->38 22 22->11 22->37 23->17 23->25 24->18 24->26 25->4 25->31 26->3 26->32 27->9 27->19 28->10 28->20 29->2 29->23 30->1 30->24 31->4 31->5 32->3 32->6 33->12 36 33->36 34->11 35 34->35 35->43 36->44 37->13 37->19 38->14 38->20 39->7 40->8 41->16 41->28 42->15 42->27 43->22 43->29 44->21 44->30 45->7 45->21 46->8 46->22
Degree $46$
Order $5.170\times 10^{22}$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2.S_{23}$

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, 45);
 
Copy content sage:G = TransitiveGroup(46, 45)
 
Copy content oscar:G = transitive_group(46, 45)
 
Copy content gap:G := TransitiveGroup(46, 45);
 

Group invariants

Abstract group:  $C_2.S_{23}$
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:  $51704033477769953280000=2^{20} \cdot 3^{9} \cdot 5^{4} \cdot 7^{3} \cdot 11^{2} \cdot 13 \cdot 17 \cdot 19 \cdot 23$
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:  no
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 $t$:  $45$
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:  $(1,40)(2,39)(3,13,29,23,17,25,31,4,14,30,24,18,26,32)(5,15,45,7,41,28,10,33,36,44,21,12,38,20)(6,16,46,8,42,27,9,34,35,43,22,11,37,19)$, $(1,44,30)(2,43,29)(3,10,27,19,17,24,26)(4,9,28,20,18,23,25)(5,40,8,41,16,45,21,38,14,34,11,31)(6,39,7,42,15,46,22,37,13,33,12,32)$
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
$4$:  $C_2^2$
$25852016738884976640000$:  $S_{23}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 23: $S_{23}$

Low degree siblings

46T45

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

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