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Properties

Label 9T6
9T6 1 2 1->2 4 1->4 3 2->3 8 2->8 3->4 5 4->5 7 4->7 5->2 6 5->6 6->7 7->1 7->8 8->5 9 8->9 9->1
Degree $9$
Order $27$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_9:C_3$

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, 6);
 
Copy content sage:G = TransitiveGroup(9, 6)
 
Copy content oscar:G = transitive_group(9, 6)
 
Copy content gap:G := TransitiveGroup(9, 6);
 

Group invariants

Abstract group:  $C_9:C_3$
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:  $27=3^{3}$
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:  $2$
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 $t$:  $6$
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:   $1/3[3^{3}]3$
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);
 
$\card{\Aut(F/K)}$:  $3$
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:  $(1,4,7)(2,8,5)$, $(1,2,3,4,5,6,7,8,9)$
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)
$3$:  $C_3$ x 4
$9$:  $C_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

27T5

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{9}$ $1$ $1$ $0$ $()$
3A1 $3^{3}$ $1$ $3$ $6$ $(1,4,7)(2,5,8)(3,6,9)$
3A-1 $3^{3}$ $1$ $3$ $6$ $(1,7,4)(2,8,5)(3,9,6)$
3B1 $3^{2},1^{3}$ $3$ $3$ $4$ $(1,7,4)(2,5,8)$
3B-1 $3^{2},1^{3}$ $3$ $3$ $4$ $(1,4,7)(2,8,5)$
9A1 $9$ $3$ $9$ $8$ $(1,5,6,4,8,9,7,2,3)$
9A-1 $9$ $3$ $9$ $8$ $(1,6,8,7,3,5,4,9,2)$
9B1 $9$ $3$ $9$ $8$ $(1,2,6,4,5,9,7,8,3)$
9B-1 $9$ $3$ $9$ $8$ $(1,9,2,7,6,8,4,3,5)$
9C1 $9$ $3$ $9$ $8$ $(1,3,5,7,9,2,4,6,8)$
9C-1 $9$ $3$ $9$ $8$ $(1,8,6,4,2,9,7,5,3)$

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 3A1 3A-1 3B1 3B-1 9A1 9A-1 9B1 9B-1 9C1 9C-1
Size 1 1 1 3 3 3 3 3 3 3 3
3 P 1A 3A-1 3A1 3B-1 3B1 9A-1 9A1 9B-1 9B1 9C-1 9C1
Type
27.4.1a R 1 1 1 1 1 1 1 1 1 1 1
27.4.1b1 C 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1
27.4.1b2 C 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1
27.4.1c1 C 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1
27.4.1c2 C 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3
27.4.1d1 C 1 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3
27.4.1d2 C 1 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1
27.4.1e1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
27.4.1e2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
27.4.3a1 C 3 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0
27.4.3a2 C 3 3ζ3 3ζ3−1 0 0 0 0 0 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

$f_{ 1 } =$ $\left(60466176 t^{8} - 26873856 t^{6} + 4478976 t^{4} - 331776 t^{2} + 9216\right) x^{9} + \left(-41150592 t^{10} - 112627584 t^{8} + 30751488 t^{6} - 4955904 t^{4} + 611712 t^{2} - 31104\right) x^{7} + \left(-22861440 t^{10} - 62570880 t^{8} + 17084160 t^{6} - 2753280 t^{4} + 339840 t^{2} - 17280\right) x^{6} + \left(7001316 t^{12} + 38007144 t^{10} + 50494428 t^{8} - 808272 t^{6} + 2196828 t^{4} - 36504 t^{2} + 23652\right) x^{5} + \left(7779240 t^{12} + 46040400 t^{10} + 66533400 t^{8} - 3745440 t^{6} + 2899800 t^{4} - 97200 t^{2} + 29160\right) x^{4} + \left(3306177 t^{12} + 19567170 t^{10} + 28276695 t^{8} - 1591812 t^{6} + 1232415 t^{4} - 41310 t^{2} + 12393\right) x^{3} + \left(669879 t^{12} + 3964590 t^{10} + 5729265 t^{8} - 322524 t^{6} + 249705 t^{4} - 8370 t^{2} + 2511\right) x^{2} + \left(64827 t^{12} + 383670 t^{10} + 554445 t^{8} - 31212 t^{6} + 24165 t^{4} - 810 t^{2} + 243\right) x + \left(2401 t^{12} + 14210 t^{10} + 20535 t^{8} - 1156 t^{6} + 895 t^{4} - 30 t^{2} + 9\right)$ Copy content Toggle raw display