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Properties

Label 18T28
18T28 1 8 1->8 12 1->12 2 7 2->7 11 2->11 3 9 3->9 18 3->18 4 10 4->10 17 4->17 5 5->12 14 5->14 6 6->11 13 6->13 7->1 7->13 8->2 8->14 9->7 16 9->16 10->8 15 10->15 11->4 11->17 12->3 12->18 13->2 13->9 14->1 14->10 15->4 15->16 16->3 17->5 17->6 18->5 18->6
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $F_9$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(18, 28);
 
Copy content sage:G = TransitiveGroup(18, 28)
 
Copy content oscar:G = transitive_group(18, 28)
 
Copy content gap:G := TransitiveGroup(18, 28);
 

Group invariants

Abstract group:  $F_9$
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:  $72=2^{3} \cdot 3^{2}$
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 $n$:  $18$
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 $t$:  $28$
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);
 
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);
 
$\card{\Aut(F/K)}$:  $2$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(18).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(18), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(18), G));
 
Generators:  $(1,8,14)(2,7,13)(3,9,16)(4,10,15)(5,12,18)(6,11,17)$, $(1,12,3,18,6,13,9,7)(2,11,4,17,5,14,10,8)(15,16)$
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)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $C_3^2:C_8$

Low degree siblings

9T15, 12T46, 24T81, 36T49

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{18}$ $1$ $1$ $0$ $()$
2A $2^{8},1^{2}$ $9$ $2$ $8$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,17)(10,18)(11,16)(12,15)$
3A $3^{6}$ $8$ $3$ $12$ $( 1,17, 3)( 2,18, 4)( 5,10, 7)( 6, 9, 8)(11,16,14)(12,15,13)$
4A1 $4^{4},1^{2}$ $9$ $4$ $12$ $( 1, 9, 8,17)( 2,10, 7,18)( 3,11, 6,16)( 4,12, 5,15)$
4A-1 $4^{4},1^{2}$ $9$ $4$ $12$ $( 1,17, 8, 9)( 2,18, 7,10)( 3,16, 6,11)( 4,15, 5,12)$
8A1 $8^{2},2$ $9$ $8$ $15$ $( 1, 4, 9,12, 8, 5,17,15)( 2, 3,10,11, 7, 6,18,16)(13,14)$
8A-1 $8^{2},2$ $9$ $8$ $15$ $( 1,15,17, 5, 8,12, 9, 4)( 2,16,18, 6, 7,11,10, 3)(13,14)$
8A3 $8^{2},2$ $9$ $8$ $15$ $( 1,12,17, 4, 8,15, 9, 5)( 2,11,18, 3, 7,16,10, 6)(13,14)$
8A-3 $8^{2},2$ $9$ $8$ $15$ $( 1, 5, 9,15, 8, 4,17,12)( 2, 6,10,16, 7, 3,18,11)(13,14)$

Malle's constant $a(G)$:     $1/8$

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 4A1 4A-1 8A1 8A-1 8A3 8A-3
Size 1 9 8 9 9 9 9 9 9
2 P 1A 1A 3A 2A 2A 4A1 4A-1 4A-1 4A1
3 P 1A 2A 1A 4A-1 4A1 8A3 8A-3 8A1 8A-1
Type
72.39.1a R 1 1 1 1 1 1 1 1 1
72.39.1b R 1 1 1 1 1 −1 −1 −1 −1
72.39.1c1 C 1 1 1 −1 −1 −i i i −i
72.39.1c2 C 1 1 1 −1 −1 i −i −i i
72.39.1d1 C 1 −1 1 −ζ82 ζ82 −ζ8 ζ83 −ζ83 ζ8
72.39.1d2 C 1 −1 1 ζ82 −ζ82 ζ83 −ζ8 ζ8 −ζ83
72.39.1d3 C 1 −1 1 −ζ82 ζ82 ζ8 −ζ83 ζ83 −ζ8
72.39.1d4 C 1 −1 1 ζ82 −ζ82 −ζ83 ζ8 −ζ8 ζ83
72.39.8a R 8 0 −1 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

Data not computed