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Properties

Label 18T13
18T13 1 7 1->7 11 1->11 18 1->18 2 8 2->8 12 2->12 17 2->17 3 6 3->6 14 3->14 15 3->15 4 5 4->5 13 4->13 16 4->16 5->14 5->15 6->13 6->16 7->12 7->17 8->11 8->18 9 9->1 10 9->10 9->17 10->2 10->18 11->3 11->16 12->4 12->15 13->6 13->14 14->5 15->8 16->7 17->9 18->10
Degree $18$
Order $36$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_{18}$

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, 13);
 
Copy content sage:G = TransitiveGroup(18, 13)
 
Copy content oscar:G = transitive_group(18, 13)
 
Copy content gap:G := TransitiveGroup(18, 13);
 

Group invariants

Abstract group:  $D_{18}$
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:  $36=2^{2} \cdot 3^{2}$
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 $t$:  $13$
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:  $(1,11,3,14,5,15,8,18,10,2,12,4,13,6,16,7,17,9)$, $(1,18)(2,17)(3,15)(4,16)(5,14)(6,13)(7,12)(8,11)(9,10)$, $(1,7)(2,8)(3,6)(4,5)(9,17)(10,18)(11,16)(12,15)(13,14)$
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
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$18$:  $D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: $D_{9}$

Low degree siblings

18T13, 36T10

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 $2^{9}$ $1$ $2$ $9$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
2B $2^{8},1^{2}$ $9$ $2$ $8$ $( 1,17)( 2,18)( 3,16)( 4,15)( 5,13)( 6,14)( 7,11)( 8,12)$
2C $2^{9}$ $9$ $2$ $9$ $( 1,18)( 2,17)( 3,15)( 4,16)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)$
3A $3^{6}$ $2$ $3$ $12$ $( 1, 8,13)( 2, 7,14)( 3,10,16)( 4, 9,15)( 5,12,17)( 6,11,18)$
6A $6^{3}$ $2$ $6$ $15$ $( 1,14, 8, 2,13, 7)( 3,15,10, 4,16, 9)( 5,18,12, 6,17,11)$
9A1 $9^{2}$ $2$ $9$ $16$ $( 1, 3, 5, 8,10,12,13,16,17)( 2, 4, 6, 7, 9,11,14,15,18)$
9A2 $9^{2}$ $2$ $9$ $16$ $( 1, 5,10,13,17, 3, 8,12,16)( 2, 6, 9,14,18, 4, 7,11,15)$
9A4 $9^{2}$ $2$ $9$ $16$ $( 1,10,17, 8,16, 5,13, 3,12)( 2, 9,18, 7,15, 6,14, 4,11)$
18A1 $18$ $2$ $18$ $17$ $( 1,11, 3,14, 5,15, 8,18,10, 2,12, 4,13, 6,16, 7,17, 9)$
18A5 $18$ $2$ $18$ $17$ $( 1, 6,10,14,17, 4, 8,11,16, 2, 5, 9,13,18, 3, 7,12,15)$
18A7 $18$ $2$ $18$ $17$ $( 1,18,16,14,12, 9, 8, 6, 3, 2,17,15,13,11,10, 7, 5, 4)$

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 2B 2C 3A 6A 9A1 9A2 9A4 18A1 18A5 18A7
Size 1 1 9 9 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 3A 9A2 9A4 9A1 9A1 9A4 9A2
3 P 1A 2A 2B 2C 1A 2A 3A 3A 3A 6A 6A 6A
Type
36.4.1a R 1 1 1 1 1 1 1 1 1 1 1 1
36.4.1b R 1 −1 −1 1 1 −1 1 1 1 −1 −1 −1
36.4.1c R 1 −1 1 −1 1 −1 1 1 1 −1 −1 −1
36.4.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1
36.4.2a R 2 2 0 0 2 2 −1 −1 −1 −1 −1 −1
36.4.2b R 2 −2 0 0 2 −2 −1 −1 −1 1 1 1
36.4.2c1 R 2 2 0 0 −1 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94
36.4.2c2 R 2 2 0 0 −1 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92
36.4.2c3 R 2 2 0 0 −1 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9
36.4.2d1 R 2 −2 0 0 −1 1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94
36.4.2d2 R 2 −2 0 0 −1 1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92
36.4.2d3 R 2 −2 0 0 −1 1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9

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