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Properties

Label 18T2
18T2 1 7 1->7 16 1->16 2 8 2->8 15 2->15 3 9 3->9 11 3->11 4 10 4->10 12 4->12 5 5->2 5->11 6 6->1 6->12 7->3 13 7->13 8->4 14 8->14 9->15 17 9->17 10->16 18 10->18 11->8 11->17 12->7 12->18 13->2 13->9 14->1 14->10 15->4 15->6 16->3 16->5 17->6 17->14 18->5 18->13
Degree $18$
Order $18$
Cyclic no
Abelian yes
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_6 \times C_3$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(18, 2);
 
Copy content sage:G = TransitiveGroup(18, 2)
 
Copy content oscar:G = transitive_group(18, 2)
 
Copy content gap:G := TransitiveGroup(18, 2);
 

Group invariants

Abstract group:  $C_6 \times C_3$
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:  $18=2 \cdot 3^{2}$
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:  yes
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:  $1$
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 $t$:  $2$
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);
 
$\card{\Aut(F/K)}$:  $18$
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:  $(1,7,13,2,8,14)(3,9,15,4,10,16)(5,11,17,6,12,18)$, $(1,16,5,2,15,6)(3,11,8,4,12,7)(9,17,14,10,18,13)$
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
$9$:  $C_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$ x 4

Degree 6: $C_6$ x 4

Degree 9: $C_3^2$

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 $3^{6}$ $1$ $3$ $12$ $( 1,13, 8)( 2,14, 7)( 3,15,10)( 4,16, 9)( 5,17,12)( 6,18,11)$
3A-1 $3^{6}$ $1$ $3$ $12$ $( 1, 8,13)( 2, 7,14)( 3,10,15)( 4, 9,16)( 5,12,17)( 6,11,18)$
3B1 $3^{6}$ $1$ $3$ $12$ $( 1,17, 3)( 2,18, 4)( 5,10, 8)( 6, 9, 7)(11,16,14)(12,15,13)$
3B-1 $3^{6}$ $1$ $3$ $12$ $( 1, 3,17)( 2, 4,18)( 5, 8,10)( 6, 7, 9)(11,14,16)(12,13,15)$
3C1 $3^{6}$ $1$ $3$ $12$ $( 1,10,12)( 2, 9,11)( 3, 5,13)( 4, 6,14)( 7,16,18)( 8,15,17)$
3C-1 $3^{6}$ $1$ $3$ $12$ $( 1,12,10)( 2,11, 9)( 3,13, 5)( 4,14, 6)( 7,18,16)( 8,17,15)$
3D1 $3^{6}$ $1$ $3$ $12$ $( 1, 5,15)( 2, 6,16)( 3, 8,12)( 4, 7,11)( 9,14,18)(10,13,17)$
3D-1 $3^{6}$ $1$ $3$ $12$ $( 1,15, 5)( 2,16, 6)( 3,12, 8)( 4,11, 7)( 9,18,14)(10,17,13)$
6A1 $6^{3}$ $1$ $6$ $15$ $( 1, 7,13, 2, 8,14)( 3, 9,15, 4,10,16)( 5,11,17, 6,12,18)$
6A-1 $6^{3}$ $1$ $6$ $15$ $( 1,14, 8, 2,13, 7)( 3,16,10, 4,15, 9)( 5,18,12, 6,17,11)$
6B1 $6^{3}$ $1$ $6$ $15$ $( 1, 4,17, 2, 3,18)( 5, 7,10, 6, 8, 9)(11,13,16,12,14,15)$
6B-1 $6^{3}$ $1$ $6$ $15$ $( 1,18, 3, 2,17, 4)( 5, 9, 8, 6,10, 7)(11,15,14,12,16,13)$
6C1 $6^{3}$ $1$ $6$ $15$ $( 1,11,10, 2,12, 9)( 3,14, 5, 4,13, 6)( 7,17,16, 8,18,15)$
6C-1 $6^{3}$ $1$ $6$ $15$ $( 1, 9,12, 2,10,11)( 3, 6,13, 4, 5,14)( 7,15,18, 8,16,17)$
6D1 $6^{3}$ $1$ $6$ $15$ $( 1,16, 5, 2,15, 6)( 3,11, 8, 4,12, 7)( 9,17,14,10,18,13)$
6D-1 $6^{3}$ $1$ $6$ $15$ $( 1, 6,15, 2, 5,16)( 3, 7,12, 4, 8,11)( 9,13,18,10,14,17)$

Malle's constant $a(G)$:     $1/9$

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 3C1 3C-1 3D1 3D-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3C-1 3C1 3D-1 3D1 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A
Type
18.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
18.5.1b R 1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
18.5.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1
18.5.1c2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1
18.5.1d1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1
18.5.1d2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3
18.5.1e1 C 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3
18.5.1e2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1
18.5.1f1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
18.5.1f2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
18.5.1g1 C 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 −1
18.5.1g2 C 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 −1
18.5.1h1 C 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −1 −1 −ζ3 −ζ3−1
18.5.1h2 C 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −1 −1 −ζ3−1 −ζ3
18.5.1i1 C 1 −1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
18.5.1i2 C 1 −1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
18.5.1j1 C 1 −1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
18.5.1j2 C 1 −1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −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