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Properties

Label 9T9
9T9 1 2 1->2 4 1->4 8 1->8 2->4 5 2->5 9 2->9 3 3->4 3->5 6 3->6 4->1 4->5 7 4->7 5->3 5->6 5->8 6->7 6->7 6->9 7->1 7->3 7->8 8->2 8->2 8->6 9->1 9->3
Degree $9$
Order $36$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive yes
$p$-group no
Group: $C_3^2:C_4$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(9, 9);
 
Copy content sage:G = TransitiveGroup(9, 9)
 
Copy content oscar:G = transitive_group(9, 9)
 
Copy content gap:G := TransitiveGroup(9, 9);
 

Group invariants

Abstract group:  $C_3^2:C_4$
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:  $36=2^{2} \cdot 3^{2}$
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 $n$:  $9$
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 $t$:  $9$
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:   $E(9):4$
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:  yes
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(9).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(9), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(9), G));
 
Generators:  $(1,2,9)(3,4,5)(6,7,8)$, $(1,8,2,4)(3,5,6,7)$, $(1,4,7)(2,5,8)(3,6,9)$
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)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

6T10 x 2, 12T17 x 2, 18T10, 36T14

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{9}$ $1$ $1$ $0$ $()$
2A $2^{4},1$ $9$ $2$ $4$ $(1,3)(2,5)(4,9)(6,7)$
3A $3^{3}$ $4$ $3$ $6$ $(1,6,5)(2,7,3)(4,9,8)$
3B $3^{3}$ $4$ $3$ $6$ $(1,8,3)(2,6,4)(5,9,7)$
4A1 $4^{2},1$ $9$ $4$ $6$ $(1,5,3,2)(4,7,9,6)$
4A-1 $4^{2},1$ $9$ $4$ $6$ $(1,2,3,5)(4,6,9,7)$

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 3B 4A1 4A-1
Size 1 9 4 4 9 9
2 P 1A 1A 3A 3B 2A 2A
3 P 1A 2A 1A 1A 4A-1 4A1
Type
36.9.1a R 1 1 1 1 1 1
36.9.1b R 1 1 1 1 −1 −1
36.9.1c1 C 1 −1 1 1 −i i
36.9.1c2 C 1 −1 1 1 i −i
36.9.4a R 4 0 −2 1 0 0
36.9.4b R 4 0 1 −2 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

$f_{ 1 } =$ $x^{9} - 18 x^{7} - 12 x^{6} + 81 x^{5} + 36 x^{4} + \left(3 t^{2} + 264\right) x^{3} + 144 x - 64$ Copy content Toggle raw display