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Properties

Label 9T18
9T18 1 2 1->2 1->2 1->2 4 1->4 5 2->5 9 2->9 3 3->4 3->4 3->5 6 3->6 3->6 4->5 4->5 7 4->7 8 4->8 5->3 5->3 5->7 5->8 6->7 6->7 6->8 6->9 7->1 7->6 7->8 8->2 8->6 8->7 9->1 9->3
Degree $9$
Order $108$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^2 : D_{6} $

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(9, 18);
 
Copy content sage:G = TransitiveGroup(9, 18)
 
Copy content oscar:G = transitive_group(9, 18)
 
Copy content gap:G := TransitiveGroup(9, 18);
 

Group invariants

Abstract group:  $C_3^2 : D_{6} $
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:  $108=2^{2} \cdot 3^{3}$
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 $t$:  $18$
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:   $E(9):D_{12}=[3^{2}:2]S(3)=[1/2.S(3)^{2}]S(3)$
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(9).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(9), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(9), G));
 
Generators:  $(1,2)(3,5)(6,7)$, $(1,2,9)(3,4,5)(6,7,8)$, $(1,2)(3,6)(4,8)(5,7)$, $(3,4,5)(6,8,7)$, $(1,4,7)(2,5,8)(3,6,9)$
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)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

9T18, 18T51 x 2, 18T55 x 2, 18T56, 18T57 x 2, 27T29, 36T87 x 2, 36T90

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 $2^{4},1$ $9$ $2$ $4$ $(1,8)(2,7)(4,5)(6,9)$
2B $2^{3},1^{3}$ $9$ $2$ $3$ $(1,9)(3,4)(6,7)$
2C $2^{3},1^{3}$ $9$ $2$ $3$ $(1,6)(2,7)(8,9)$
3A $3^{3}$ $2$ $3$ $6$ $(1,9,2)(3,5,4)(6,8,7)$
3B $3^{2},1^{3}$ $6$ $3$ $4$ $(1,9,2)(6,7,8)$
3C $3^{3}$ $6$ $3$ $6$ $(1,7,4)(2,8,5)(3,9,6)$
3D $3^{3}$ $12$ $3$ $6$ $(1,4,6)(2,5,7)(3,8,9)$
6A $6,2,1$ $18$ $6$ $6$ $(1,7,9,8,2,6)(4,5)$
6B $6,3$ $18$ $6$ $7$ $(1,3,7,9,4,6)(2,5,8)$
6C $6,3$ $18$ $6$ $7$ $(1,7,9,6,2,8)(3,4,5)$

Malle's constant $a(G)$:     $1/3$

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 2B 2C 3A 3B 3C 3D 6A 6B 6C
Size 1 9 9 9 2 6 6 12 18 18 18
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3B 3C 3A
3 P 1A 2A 2B 2C 1A 1A 1A 1A 2A 2B 2C
Type
108.17.1a R 1 1 1 1 1 1 1 1 1 1 1
108.17.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1
108.17.1c R 1 −1 1 −1 1 1 1 1 −1 −1 1
108.17.1d R 1 1 −1 −1 1 1 1 1 −1 1 −1
108.17.2a R 2 0 2 0 2 2 −1 −1 0 0 −1
108.17.2b R 2 2 0 0 2 −1 2 −1 0 −1 0
108.17.2c R 2 −2 0 0 2 −1 2 −1 0 1 0
108.17.2d R 2 0 −2 0 2 2 −1 −1 0 0 1
108.17.4a R 4 0 0 0 4 −2 −2 1 0 0 0
108.17.6a R 6 0 0 2 −3 0 0 0 −1 0 0
108.17.6b R 6 0 0 −2 −3 0 0 0 1 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

$f_{ 1 } =$ $4 x^{9} + t x^{7} + 12 x^{6} + 2 t x^{4} - 15 x^{3} + t x + 4$ Copy content Toggle raw display