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Properties

Label 18T24
18T24 1 2 1->2 8 1->8 7 2->7 3 4 3->4 10 3->10 9 4->9 5 5->7 11 5->11 12 5->12 6 6->8 6->11 6->12 7->9 13 7->13 14 7->14 8->10 8->13 8->14 9->5 15 9->15 16 9->16 10->6 10->15 10->16 11->15 17 11->17 12->16 18 12->18 13->1 13->11 14->2 14->12 15->3 15->13 16->4 16->14 17->6 17->18 18->5
Degree $18$
Order $54$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^2:S_3$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(18, 24);
 
Copy content sage:G = TransitiveGroup(18, 24)
 
Copy content oscar:G = transitive_group(18, 24)
 
Copy content gap:G := TransitiveGroup(18, 24);
 

Group invariants

Abstract group:  $C_3^2:S_3$
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:  $54=2 \cdot 3^{3}$
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:   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 $t$:  $24$
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);
 
$\card{\Aut(F/K)}$:  $6$
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:  $(5,7,9)(6,8,10)(11,15,13)(12,16,14)$, $(1,8,13)(2,7,14)(3,10,15)(4,9,16)(5,12,18)(6,11,17)$, $(1,2)(3,4)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(17,18)$
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)
$2$:  $C_2$
$6$:  $S_3$ x 4
$18$:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: $(C_3^2:C_3):C_2$

Low degree siblings

9T12 x 4, 18T24 x 3, 27T6

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

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 3A1 3A-1 3B 3C 3D 3E 6A1 6A-1
Size 1 9 1 1 6 6 6 6 9 9
2 P 1A 1A 3A-1 3A1 3B 3C 3D 3E 3A1 3A-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 2A 2A
Type
54.8.1a R 1 1 1 1 1 1 1 1 1 1
54.8.1b R 1 −1 1 1 1 1 1 1 −1 −1
54.8.2a R 2 0 2 2 −1 −1 −1 2 0 0
54.8.2b R 2 0 2 2 −1 −1 2 −1 0 0
54.8.2c R 2 0 2 2 −1 2 −1 −1 0 0
54.8.2d R 2 0 2 2 2 −1 −1 −1 0 0
54.8.3a1 C 3 1 3ζ3−1 3ζ3 0 0 0 0 ζ3 ζ3−1
54.8.3a2 C 3 1 3ζ3 3ζ3−1 0 0 0 0 ζ3−1 ζ3
54.8.3b1 C 3 −1 3ζ3−1 3ζ3 0 0 0 0 −ζ3 −ζ3−1
54.8.3b2 C 3 −1 3ζ3 3ζ3−1 0 0 0 0 −ζ3−1 −ζ3

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