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Properties

Label 9T32
9T32 1 2 1->2 9 1->9 3 2->3 4 2->4 2->4 5 2->5 3->5 6 3->6 3->6 4->3 4->5 4->6 7 4->7 5->1 5->7 6->2 6->7 6->7 7->3 7->5 8 8->9
Degree $9$
Order $1512$
Cyclic no
Abelian no
Solvable no
Transitivity $3$
Primitive yes
$p$-group no
Group: $\mathrm{P}\Gamma\mathrm{L}(2,8)$

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, 32);
 
Copy content sage:G = TransitiveGroup(9, 32)
 
Copy content oscar:G = transitive_group(9, 32)
 
Copy content gap:G := TransitiveGroup(9, 32);
 

Group invariants

Abstract group:  $\mathrm{P}\Gamma\mathrm{L}(2,8)$
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:  $1512=2^{3} \cdot 3^{3} \cdot 7$
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 $n$:  $9$
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 $t$:  $32$
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:   $L(9):3=P|L(2,8)$
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:  3
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:  $(2,5)(3,6)(4,7)(8,9)$, $(1,9)(2,3)(4,5)(6,7)$, $(1,2,4,3,6,7,5)$, $(2,4,6)(3,5,7)$
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)
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

27T391, 28T165, 36T2342

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 $2^{4},1$ $63$ $2$ $4$ $(2,8)(3,7)(4,6)(5,9)$
3A $3^{3}$ $56$ $3$ $6$ $(1,3,8)(2,5,6)(4,7,9)$
3B1 $3^{2},1^{3}$ $84$ $3$ $4$ $(2,6,7)(3,8,4)$
3B-1 $3^{2},1^{3}$ $84$ $3$ $4$ $(2,7,6)(3,4,8)$
6A1 $6,2,1$ $252$ $6$ $6$ $(2,3,6,8,7,4)(5,9)$
6A-1 $6,2,1$ $252$ $6$ $6$ $(2,4,7,8,6,3)(5,9)$
7A $7,1^{2}$ $216$ $7$ $6$ $(1,6,7,9,4,8,5)$
9A $9$ $168$ $9$ $8$ $(1,5,7,3,6,9,8,2,4)$
9B1 $9$ $168$ $9$ $8$ $(1,8,7,5,9,3,6,2,4)$
9B-1 $9$ $168$ $9$ $8$ $(1,4,2,6,3,9,5,7,8)$

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 3B1 3B-1 6A1 6A-1 7A 9A 9B1 9B-1
Size 1 63 56 84 84 252 252 216 168 168 168
2 P 1A 1A 3A 3B-1 3B1 3B1 3B-1 7A 9A 9B-1 9B1
3 P 1A 2A 1A 1A 1A 2A 2A 7A 3A 3A 3A
7 P 1A 2A 3A 3B1 3B-1 6A1 6A-1 1A 9A 9B1 9B-1
Type
1512.779.1a R 1 1 1 1 1 1 1 1 1 1 1
1512.779.1b1 C 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 ζ3 1 ζ3−1
1512.779.1b2 C 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 ζ3−1 1 ζ3
1512.779.7a R 7 −1 −2 1 1 −1 −1 0 1 1 1
1512.779.7b1 C 7 −1 −2 ζ3−1 ζ3 −ζ3 −ζ3−1 0 ζ3 1 ζ3−1
1512.779.7b2 C 7 −1 −2 ζ3 ζ3−1 −ζ3−1 −ζ3 0 ζ3−1 1 ζ3
1512.779.8a R 8 0 −1 2 2 0 0 1 −1 −1 −1
1512.779.8b1 C 8 0 −1 2ζ3−1 2ζ3 0 0 1 −ζ3 −1 −ζ3−1
1512.779.8b2 C 8 0 −1 2ζ3 2ζ3−1 0 0 1 −ζ3−1 −1 −ζ3
1512.779.21a R 21 −3 3 0 0 0 0 0 0 0 0
1512.779.27a R 27 3 0 0 0 0 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

$f_{ 1 } =$ $x^{9} + 108 x^{7} + 216 x^{6} + 4374 x^{5} + 13608 x^{4} + 99468 x^{3}+215784 x^{2} + 998001 x + 663552 t + 810648$ Copy content Toggle raw display