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Properties

Label 12T47
12T47 1 2 1->2 4 1->4 5 1->5 2->5 6 2->6 7 2->7 10 2->10 3 3->6 3->7 12 3->12 4->5 8 4->8 4->8 11 4->11 5->8 5->10 9 6->9 6->10 7->4 7->10 7->11 7->11 8->1 8->7 8->12 9->12 10->1 10->2 10->11 11->2 11->3 11->8 12->4
Degree $12$
Order $72$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $\PSU(3,2)$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(12, 47);
 
Copy content sage:G = TransitiveGroup(12, 47)
 
Copy content oscar:G = transitive_group(12, 47)
 
Copy content gap:G := TransitiveGroup(12, 47);
 

Group invariants

Abstract group:  $\PSU(3,2)$
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:  $72=2^{3} \cdot 3^{2}$
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 $n$:  $12$
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 $t$:  $47$
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);
 
CHM label:   $[(1/3.3^{3}):2]E(4)_{4}$
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(12).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(12), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(12), G));
 
Generators:  $(1,2,5,10)(3,12)(4,11,8,7)(6,9)$, $(1,5)(2,10)(4,8)(7,11)$, $(1,4,5,8)(2,7,10,11)(3,6)(9,12)$, $(2,6,10)(3,7,11)(4,8,12)$
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
$4$:  $C_2^2$
$8$:  $Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: None

Low degree siblings

9T14, 18T35 x 3, 24T82, 36T55

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{4},1^{4}$ $9$ $2$ $4$ $( 1, 9)( 6,10)( 7,11)( 8,12)$
3A $3^{3},1^{3}$ $8$ $3$ $6$ $( 1, 9, 5)( 2, 6,10)( 3,11, 7)$
4A $4^{2},2^{2}$ $18$ $4$ $8$ $( 1, 7, 9,11)( 2, 4)( 3, 5)( 6,12,10, 8)$
4B $4^{2},2^{2}$ $18$ $4$ $8$ $( 1, 2, 5,10)( 3,12)( 4,11, 8, 7)( 6, 9)$
4C $4^{2},2^{2}$ $18$ $4$ $8$ $( 1, 4)( 2, 3,10, 7)( 5, 8, 9,12)( 6,11)$

Malle's constant $a(G)$:     $1/4$

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

1A 2A 3A 4A 4B 4C
Size 1 9 8 18 18 18
2 P 1A 1A 3A 2A 2A 2A
3 P 1A 2A 1A 4A 4B 4C
Type
72.41.1a R 1 1 1 1 1 1
72.41.1b R 1 1 1 −1 −1 1
72.41.1c R 1 1 1 −1 1 −1
72.41.1d R 1 1 1 1 −1 −1
72.41.2a S 2 −2 2 0 0 0
72.41.8a R 8 0 −1 0 0 0

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

Additional information

This is an elusive group.