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Properties

Label 18T26
18T26 1 7 1->7 8 1->8 2 2->7 2->8 3 9 3->9 10 3->10 4 4->9 4->10 5 11 5->11 12 5->12 6 6->11 6->12 16 7->16 7->16 15 8->15 8->15 18 9->18 9->18 17 10->17 10->17 13 11->13 11->13 14 12->14 12->14 13->3 13->3 14->4 14->4 15->5 15->5 16->6 16->6 17->1 17->1 18->2 18->2
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2^2:C_{18}$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(18, 26);
 
Copy content sage:G = TransitiveGroup(18, 26)
 
Copy content oscar:G = transitive_group(18, 26)
 
Copy content gap:G := TransitiveGroup(18, 26);
 

Group invariants

Abstract group:  $C_2^2:C_{18}$
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:  $72=2^{3} \cdot 3^{2}$
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 $t$:  $26$
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:  $(1,7,16,6,11,13,3,10,17)(2,8,15,5,12,14,4,9,18)$, $(1,8,15,5,11,13,3,9,18,2,7,16,6,12,14,4,10,17)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$9$:  $C_9$
$12$:  $A_4$
$18$:  $C_{18}$
$24$:  $A_4\times C_2$
$36$:  $C_2^2 : C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4\times C_2$

Degree 9: $C_9$

Low degree siblings

36T16, 36T30

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 $2^{6},1^{6}$ $3$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
2C $2^{3},1^{12}$ $3$ $2$ $3$ $(13,14)(15,16)(17,18)$
3A1 $3^{6}$ $1$ $3$ $12$ $( 1, 3, 6)( 2, 4, 5)( 7,10,11)( 8, 9,12)(13,16,17)(14,15,18)$
3A-1 $3^{6}$ $1$ $3$ $12$ $( 1, 6, 3)( 2, 5, 4)( 7,11,10)( 8,12, 9)(13,17,16)(14,18,15)$
6A1 $6^{3}$ $1$ $6$ $15$ $( 1, 4, 6, 2, 3, 5)( 7, 9,11, 8,10,12)(13,15,17,14,16,18)$
6A-1 $6^{3}$ $1$ $6$ $15$ $( 1, 5, 3, 2, 6, 4)( 7,12,10, 8,11, 9)(13,18,16,14,17,15)$
6B1 $6^{2},3^{2}$ $3$ $6$ $14$ $( 1, 5, 3, 2, 6, 4)( 7,12,10, 8,11, 9)(13,17,16)(14,18,15)$
6B-1 $6^{2},3^{2}$ $3$ $6$ $14$ $( 1, 4, 6, 2, 3, 5)( 7, 9,11, 8,10,12)(13,16,17)(14,15,18)$
6C1 $6,3^{4}$ $3$ $6$ $13$ $( 1, 3, 6)( 2, 4, 5)( 7,10,11)( 8, 9,12)(13,15,17,14,16,18)$
6C-1 $6,3^{4}$ $3$ $6$ $13$ $( 1, 6, 3)( 2, 5, 4)( 7,11,10)( 8,12, 9)(13,18,16,14,17,15)$
9A1 $9^{2}$ $4$ $9$ $16$ $( 1, 7,16, 6,11,13, 3,10,17)( 2, 8,15, 5,12,14, 4, 9,18)$
9A-1 $9^{2}$ $4$ $9$ $16$ $( 1,17,10, 3,13,11, 6,16, 7)( 2,18, 9, 4,14,12, 5,15, 8)$
9A2 $9^{2}$ $4$ $9$ $16$ $( 1,16,11, 3,17, 7, 6,13,10)( 2,15,12, 4,18, 8, 5,14, 9)$
9A-2 $9^{2}$ $4$ $9$ $16$ $( 1,10,13, 6, 7,17, 3,11,16)( 2, 9,14, 5, 8,18, 4,12,15)$
9A4 $9^{2}$ $4$ $9$ $16$ $( 1,11,17, 6,10,16, 3, 7,13)( 2,12,18, 5, 9,15, 4, 8,14)$
9A-4 $9^{2}$ $4$ $9$ $16$ $( 1,13, 7, 3,16,10, 6,17,11)( 2,14, 8, 4,15, 9, 5,18,12)$
18A1 $18$ $4$ $18$ $17$ $( 1,14, 7, 4,16, 9, 6,18,11, 2,13, 8, 3,15,10, 5,17,12)$
18A-1 $18$ $4$ $18$ $17$ $( 1,12,17, 5,10,15, 3, 8,13, 2,11,18, 6, 9,16, 4, 7,14)$
18A5 $18$ $4$ $18$ $17$ $( 1, 9,13, 5, 7,18, 3,12,16, 2,10,14, 6, 8,17, 4,11,15)$
18A-5 $18$ $4$ $18$ $17$ $( 1,15,11, 4,17, 8, 6,14,10, 2,16,12, 3,18, 7, 5,13, 9)$
18A7 $18$ $4$ $18$ $17$ $( 1,18,10, 4,13,12, 6,15, 7, 2,17, 9, 3,14,11, 5,16, 8)$
18A-7 $18$ $4$ $18$ $17$ $( 1, 8,16, 5,11,14, 3, 9,17, 2, 7,15, 6,12,13, 4,10,18)$

Malle's constant $a(G)$:     $1/3$

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 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 18A1 18A-1 18A5 18A-5 18A7 18A-7
Size 1 1 3 3 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 3A-1 3A1 3A-1 3A1 3A1 3A-1 3A-1 3A1 9A2 9A-2 9A4 9A-4 9A-1 9A1 9A1 9A-1 9A-4 9A4 9A-2 9A2
3 P 1A 2A 2B 2C 1A 1A 2A 2A 2B 2B 2C 2C 3A-1 3A1 3A1 3A-1 3A-1 3A1 6A1 6A-1 6A-1 6A1 6A1 6A-1
Type
72.16.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.16.1b R 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
72.16.1c1 C 1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
72.16.1c2 C 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
72.16.1d1 C 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
72.16.1d2 C 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
72.16.1e1 C 1 1 1 1 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ94 ζ92 ζ9−1 ζ9 ζ9−4 ζ9−2 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2
72.16.1e2 C 1 1 1 1 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ9−4 ζ9−2 ζ9 ζ9−1 ζ94 ζ92 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92
72.16.1e3 C 1 1 1 1 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−2 ζ9−1 ζ9−4 ζ94 ζ92 ζ9 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9
72.16.1e4 C 1 1 1 1 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ92 ζ9 ζ94 ζ9−4 ζ9−2 ζ9−1 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1
72.16.1e5 C 1 1 1 1 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9 ζ9−4 ζ92 ζ9−2 ζ9−1 ζ94 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94
72.16.1e6 C 1 1 1 1 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ9−1 ζ94 ζ9−2 ζ92 ζ9 ζ9−4 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4
72.16.1f1 C 1 −1 1 −1 ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 ζ94 ζ92 ζ9−1 ζ9 ζ9−4 ζ9−2 −ζ9−4 −ζ94 −ζ9 −ζ9−1 −ζ92 −ζ9−2
72.16.1f2 C 1 −1 1 −1 ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 ζ9−4 ζ9−2 ζ9 ζ9−1 ζ94 ζ92 −ζ94 −ζ9−4 −ζ9−1 −ζ9 −ζ9−2 −ζ92
72.16.1f3 C 1 −1 1 −1 ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 ζ9−2 ζ9−1 ζ9−4 ζ94 ζ92 ζ9 −ζ92 −ζ9−2 −ζ94 −ζ9−4 −ζ9−1 −ζ9
72.16.1f4 C 1 −1 1 −1 ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 ζ92 ζ9 ζ94 ζ9−4 ζ9−2 ζ9−1 −ζ9−2 −ζ92 −ζ9−4 −ζ94 −ζ9 −ζ9−1
72.16.1f5 C 1 −1 1 −1 ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 ζ9 ζ9−4 ζ92 ζ9−2 ζ9−1 ζ94 −ζ9−1 −ζ9 −ζ9−2 −ζ92 −ζ9−4 −ζ94
72.16.1f6 C 1 −1 1 −1 ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 ζ9−1 ζ94 ζ9−2 ζ92 ζ9 ζ9−4 −ζ9 −ζ9−1 −ζ92 −ζ9−2 −ζ94 −ζ9−4
72.16.3a R 3 3 −1 −1 3 3 3 3 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3b R 3 −3 −1 1 3 3 −3 −3 −1 1 1 −1 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3c1 C 3 3 −1 −1 3ζ3−1 3ζ3 3ζ3−1 3ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3c2 C 3 3 −1 −1 3ζ3 3ζ3−1 3ζ3 3ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3d1 C 3 −3 −1 1 3ζ3−1 3ζ3 −3ζ3−1 −3ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3d2 C 3 −3 −1 1 3ζ3 3ζ3−1 −3ζ3 −3ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 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