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Properties

Label 18T14
18T14 1 3 1->3 17 1->17 2 4 2->4 18 2->18 3->2 11 3->11 4->1 12 4->12 5 5->1 5->4 6 6->2 6->3 7 7->5 9 7->9 8 8->6 10 8->10 9->8 9->17 10->7 10->18 11->8 11->9 12->7 12->10 13 13->11 16 13->16 14 14->12 15 14->15 15->5 15->13 16->6 16->14 17->13 17->15 18->14 18->16
Degree $18$
Order $54$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_9:C_6$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(18, 14);
 
Copy content sage:G = TransitiveGroup(18, 14)
 
Copy content oscar:G = transitive_group(18, 14)
 
Copy content gap:G := TransitiveGroup(18, 14);
 

Group invariants

Abstract group:  $C_9:C_6$
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:  $2$
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 $n$:  $18$
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 $t$:  $14$
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:  $(1,17,15,13,11,9,8,6,3,2,18,16,14,12,10,7,5,4)$, $(1,3,11,8,10,18,14,15,5)(2,4,12,7,9,17,13,16,6)$
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)
$2$:  $C_2$
$3$:  $C_3$ x 4
$6$:  $C_6$ x 4
$9$:  $C_3^2$
$18$:  $C_6 \times C_3$
$27$:  $C_9:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: $C_9:C_3$

Low degree siblings

There are no siblings 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 $2^{9}$ $1$ $2$ $9$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
3A1 $3^{6}$ $1$ $3$ $12$ $( 1, 8,14)( 2, 7,13)( 3,10,15)( 4, 9,16)( 5,11,18)( 6,12,17)$
3A-1 $3^{6}$ $1$ $3$ $12$ $( 1,14, 8)( 2,13, 7)( 3,15,10)( 4,16, 9)( 5,18,11)( 6,17,12)$
3B1 $3^{4},1^{6}$ $3$ $3$ $8$ $( 1, 8,14)( 2, 7,13)( 3,15,10)( 4,16, 9)$
3B-1 $3^{4},1^{6}$ $3$ $3$ $8$ $( 1,14, 8)( 2,13, 7)( 3,10,15)( 4, 9,16)$
6A1 $6^{3}$ $1$ $6$ $15$ $( 1,13, 8, 2,14, 7)( 3,16,10, 4,15, 9)( 5,17,11, 6,18,12)$
6A-1 $6^{3}$ $1$ $6$ $15$ $( 1, 7,14, 2, 8,13)( 3, 9,15, 4,10,16)( 5,12,18, 6,11,17)$
6B1 $6^{2},2^{3}$ $3$ $6$ $13$ $( 1,13, 8, 2,14, 7)( 3, 9,15, 4,10,16)( 5, 6)(11,12)(17,18)$
6B-1 $6^{2},2^{3}$ $3$ $6$ $13$ $( 1,13, 8, 2,14, 7)( 3, 4)( 5,12,18, 6,11,17)( 9,10)(15,16)$
9A1 $9^{2}$ $3$ $9$ $16$ $( 1,15,11, 8, 3,18,14,10, 5)( 2,16,12, 7, 4,17,13, 9, 6)$
9A-1 $9^{2}$ $3$ $9$ $16$ $( 1, 5,10,14,18, 3, 8,11,15)( 2, 6, 9,13,17, 4, 7,12,16)$
9B1 $9^{2}$ $3$ $9$ $16$ $( 1, 3,11, 8,10,18,14,15, 5)( 2, 4,12, 7, 9,17,13,16, 6)$
9B-1 $9^{2}$ $3$ $9$ $16$ $( 1, 5,15,14,18,10, 8,11, 3)( 2, 6,16,13,17, 9, 7,12, 4)$
9C1 $9^{2}$ $3$ $9$ $16$ $( 1,10,11, 8,15,18,14, 3, 5)( 2, 9,12, 7,16,17,13, 4, 6)$
9C-1 $9^{2}$ $3$ $9$ $16$ $( 1, 5, 3,14,18,15, 8,11,10)( 2, 6, 4,13,17,16, 7,12, 9)$
18A1 $18$ $3$ $18$ $17$ $( 1,17,15,13,11, 9, 8, 6, 3, 2,18,16,14,12,10, 7, 5, 4)$
18A-1 $18$ $3$ $18$ $17$ $( 1, 9,18, 7,15, 6,14, 4,11, 2,10,17, 8,16, 5,13, 3,12)$
18B1 $18$ $3$ $18$ $17$ $( 1,17, 3,13,11,16, 8, 6,10, 2,18, 4,14,12,15, 7, 5, 9)$
18B-1 $18$ $3$ $18$ $17$ $( 1, 9, 5, 7,15,12,14, 4,18, 2,10, 6, 8,16,11,13, 3,17)$
18C1 $18$ $3$ $18$ $17$ $( 1,17,10,13,11, 4, 8, 6,15, 2,18, 9,14,12, 3, 7, 5,16)$
18C-1 $18$ $3$ $18$ $17$ $( 1, 9,11, 7,15,17,14, 4, 5, 2,10,12, 8,16,18,13, 3, 6)$

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 3B1 3B-1 6A1 6A-1 6B1 6B-1 9A1 9A-1 9B1 9B-1 9C1 9C-1 18A1 18A-1 18B1 18B-1 18C1 18C-1
Size 1 1 1 1 3 3 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3A1 3A-1 3B1 3B-1 9A-1 9A1 9B-1 9B1 9C-1 9C1 9A1 9A-1 9B1 9B-1 9C1 9C-1
3 P 1A 2A 1A 1A 1A 1A 2A 2A 2A 2A 3A1 3A-1 3A1 3A-1 3A1 3A-1 6A1 6A-1 6A1 6A-1 6A1 6A-1
Type
54.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
54.11.1b R 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
54.11.1c1 C 1 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1
54.11.1c2 C 1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1
54.11.1d1 C 1 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1
54.11.1d2 C 1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3
54.11.1e1 C 1 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
54.11.1e2 C 1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1
54.11.1f1 C 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
54.11.1f2 C 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
54.11.1g1 C 1 −1 1 1 ζ3−1 ζ3 −1 −1 −ζ3 −ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −1 −1
54.11.1g2 C 1 −1 1 1 ζ3 ζ3−1 −1 −1 −ζ3−1 −ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −1 −1
54.11.1h1 C 1 −1 1 1 ζ3−1 ζ3 −1 −1 −ζ3 −ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −1 −ζ3 −ζ3−1
54.11.1h2 C 1 −1 1 1 ζ3 ζ3−1 −1 −1 −ζ3−1 −ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −1 −ζ3−1 −ζ3
54.11.1i1 C 1 −1 1 1 ζ3−1 ζ3 −1 −1 −ζ3 −ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
54.11.1i2 C 1 −1 1 1 ζ3 ζ3−1 −1 −1 −ζ3−1 −ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
54.11.1j1 C 1 −1 1 1 1 1 −1 −1 −1 −1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
54.11.1j2 C 1 −1 1 1 1 1 −1 −1 −1 −1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
54.11.3a1 C 3 3 3ζ3−1 3ζ3 0 0 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
54.11.3a2 C 3 3 3ζ3 3ζ3−1 0 0 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0 0 0
54.11.3b1 C 3 −3 3ζ3−1 3ζ3 0 0 −3ζ3 −3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
54.11.3b2 C 3 −3 3ζ3 3ζ3−1 0 0 −3ζ3−1 −3ζ3 0 0 0 0 0 0 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

Data not computed