Properties

Label 9T19
9T19 1 2 1->2 1->2 4 1->4 6 1->6 3 2->3 5 2->5 9 2->9 3->4 3->5 3->6 8 3->8 4->5 4->5 7 4->7 5->2 5->3 5->8 6->4 6->7 6->7 6->9 7->1 7->1 7->8 8->2 8->6 8->7 9->1 9->3
Degree $9$
Order $144$
Cyclic no
Abelian no
Solvable yes
Transitivity $2$
Primitive yes
$p$-group no
Group: $(C_3^2:C_8):C_2$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(9, 19);
 
Copy content sage:G = TransitiveGroup(9, 19)
 
Copy content oscar:G = transitive_group(9, 19)
 
Copy content gap:G := TransitiveGroup(9, 19);
 

Group invariants

Abstract group:  $(C_3^2:C_8):C_2$
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:  $144=2^{4} \cdot 3^{2}$
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 $t$:  $19$
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:   $E(9):2D_{8}$
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:  2
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:  $(1,6,4,5,2,3,8,7)$, $(1,2)(3,5)(6,7)$, $(1,2,9)(3,4,5)(6,7,8)$, $(1,4,7)(2,5,8)(3,6,9)$
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
$4$:  $C_2^2$
$8$:  $D_{4}$
$16$:  $QD_{16}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

12T84, 18T68, 18T71, 18T73, 24T278, 24T279, 24T280, 36T171, 36T172, 36T175

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 $2^{4},1$ $9$ $2$ $4$ $(2,9)(3,8)(4,7)(5,6)$
2B $2^{3},1^{3}$ $12$ $2$ $3$ $(1,6)(2,7)(8,9)$
3A $3^{3}$ $8$ $3$ $6$ $(1,2,9)(3,4,5)(6,7,8)$
4A $4^{2},1$ $18$ $4$ $6$ $(2,5,9,6)(3,4,8,7)$
4B $4^{2},1$ $36$ $4$ $6$ $(1,6,9,4)(3,8,7,5)$
6A $6,3$ $24$ $6$ $7$ $(1,8,2,6,9,7)(3,5,4)$
8A1 $8,1$ $18$ $8$ $7$ $(2,7,5,3,9,4,6,8)$
8A-1 $8,1$ $18$ $8$ $7$ $(2,8,6,4,9,3,5,7)$

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 3A 4A 4B 6A 8A1 8A-1
Size 1 9 12 8 18 36 24 18 18
2 P 1A 1A 1A 3A 2A 2A 3A 4A 4A
3 P 1A 2A 2B 1A 4A 4B 2B 8A1 8A-1
Type
144.182.1a R 1 1 1 1 1 1 1 1 1
144.182.1b R 1 1 −1 1 1 −1 −1 1 1
144.182.1c R 1 1 −1 1 1 1 −1 −1 −1
144.182.1d R 1 1 1 1 1 −1 1 −1 −1
144.182.2a R 2 2 0 2 −2 0 0 0 0
144.182.2b1 C 2 −2 0 2 0 0 0 −ζ8−ζ83 ζ8+ζ83
144.182.2b2 C 2 −2 0 2 0 0 0 ζ8+ζ83 −ζ8−ζ83
144.182.8a R 8 0 2 −1 0 0 −1 0 0
144.182.8b R 8 0 −2 −1 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

$f_{ 1 } =$ $\left(t^{2} + 8\right) x^{9} + \left(-2 t^{3} - t^{2} - 16 t - 8\right) x^{8} + \left(t^{4} + 8 t^{2}\right) x^{7} + \left(t^{4} + 8 t^{2}\right) x^{6} + \left(t^{4} + 8 t^{2}\right) x^{5} + \left(t^{4} + 8 t^{2}\right) x^{4} + \left(t^{4} + 8 t^{2}\right) x^{3} + \left(t^{4} + 8 t^{2}\right) x^{2} + \left(t^{4} - t^{2}\right) x + \left(t^{4} + 2 t^{3} + t^{2}\right)$ Copy content Toggle raw display