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Properties

Label 18T234
18T234 1 5 1->5 15 1->15 2 6 2->6 13 2->13 3 4 3->4 14 3->14 7 4->7 18 4->18 8 5->8 16 5->16 9 6->9 17 6->17 7->2 10 7->10 8->1 12 8->12 9->3 11 9->11 10->9 10->14 11->8 11->15 12->7 12->13 13->16 14->17 15->18 16->4 16->11 17->5 17->10 18->6 18->12
Degree $18$
Order $972$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^3:S_3^2$

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, 234);
 
Copy content sage:G = TransitiveGroup(18, 234)
 
Copy content oscar:G = transitive_group(18, 234)
 
Copy content gap:G := TransitiveGroup(18, 234);
 

Group invariants

Abstract group:  $C_3^3:S_3^2$
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:  $972=2^{2} \cdot 3^{5}$
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 $t$:  $234$
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);
 
$\card{\Aut(F/K)}$:  $1$
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:  $(1,5,8)(2,6,9,3,4,7)(10,14,17)(11,15,18,12,13,16)$, $(1,15)(2,13)(3,14)(4,18,6,17,5,16)(7,10,9,11,8,12)$
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
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$ x 2, $C_6$ x 3
$12$:  $D_{6}$ x 2, $C_6\times C_2$
$18$:  $S_3\times C_3$ x 2
$36$:  $S_3^2$, $C_6\times S_3$ x 2
$54$:  $C_3^2 : C_6$ x 2
$108$:  12T70, 18T41 x 2
$324$:  18T121 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: None

Low degree siblings

18T230 x 3, 18T234 x 2, 27T269 x 3, 36T1490 x 3, 36T1494 x 3, 36T1590 x 3

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 $2^{9}$ $9$ $2$ $9$ $( 1,14)( 2,15)( 3,13)( 4,18)( 5,16)( 6,17)( 7,12)( 8,11)( 9,10)$
2B $2^{9}$ $9$ $2$ $9$ $( 1,14)( 2,13)( 3,15)( 4,18)( 5,17)( 6,16)( 7,12)( 8,10)( 9,11)$
2C $2^{6},1^{6}$ $81$ $2$ $6$ $( 1, 3)( 4, 5)( 7, 9)(10,11)(13,15)(16,17)$
3A $3^{6}$ $2$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,14,15)(16,18,17)$
3B $3^{6}$ $2$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,17,18)$
3C $3^{3},1^{9}$ $4$ $3$ $6$ $(10,11,12)(13,15,14)(16,17,18)$
3D $3^{4},1^{6}$ $6$ $3$ $8$ $( 1, 2, 3)( 4, 5, 6)(13,14,15)(16,17,18)$
3E $3^{4},1^{6}$ $6$ $3$ $8$ $( 4, 6, 5)( 7, 9, 8)(10,12,11)(16,17,18)$
3F1 $3^{6}$ $9$ $3$ $12$ $( 1, 5, 8)( 2, 4, 9)( 3, 6, 7)(10,14,17)(11,13,18)(12,15,16)$
3F-1 $3^{6}$ $9$ $3$ $12$ $( 1, 8, 5)( 2, 9, 4)( 3, 7, 6)(10,17,14)(11,18,13)(12,16,15)$
3G $3^{5},1^{3}$ $12$ $3$ $10$ $( 1, 2, 3)( 4, 5, 6)(10,12,11)(13,14,15)(16,18,17)$
3H $3^{4},1^{6}$ $12$ $3$ $8$ $( 1, 2, 3)( 7, 9, 8)(13,14,15)(16,17,18)$
3I $3^{2},1^{12}$ $12$ $3$ $4$ $(4,6,5)(7,9,8)$
3J $3^{5},1^{3}$ $12$ $3$ $10$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,14,15)$
3K $3^{4},1^{6}$ $12$ $3$ $8$ $( 1, 3, 2)( 7, 8, 9)(13,14,15)(16,17,18)$
3L1 $3^{6}$ $18$ $3$ $12$ $( 1, 8, 4)( 2, 9, 6)( 3, 7, 5)(10,18,13)(11,16,15)(12,17,14)$
3L-1 $3^{6}$ $18$ $3$ $12$ $( 1, 4, 8)( 2, 6, 9)( 3, 5, 7)(10,13,18)(11,15,16)(12,14,17)$
3M1 $3^{6}$ $18$ $3$ $12$ $( 1, 9, 5)( 2, 7, 4)( 3, 8, 6)(10,18,14)(11,16,13)(12,17,15)$
3M-1 $3^{6}$ $18$ $3$ $12$ $( 1, 5, 9)( 2, 4, 7)( 3, 6, 8)(10,14,18)(11,13,16)(12,15,17)$
3N1 $3^{6}$ $36$ $3$ $12$ $( 1, 9, 4)( 2, 7, 6)( 3, 8, 5)(10,18,15)(11,16,14)(12,17,13)$
3N-1 $3^{6}$ $36$ $3$ $12$ $( 1, 4, 9)( 2, 6, 7)( 3, 5, 8)(10,15,18)(11,14,16)(12,13,17)$
6A $6^{3}$ $18$ $6$ $15$ $( 1,13, 2,14, 3,15)( 4,16, 6,18, 5,17)( 7,10, 8,12, 9,11)$
6B $6^{3}$ $18$ $6$ $15$ $( 1,15, 2,14, 3,13)( 4,17, 6,18, 5,16)( 7,11, 8,12, 9,10)$
6C1 $6^{3}$ $27$ $6$ $15$ $( 1,10, 5,14, 8,17)( 2,11, 4,13, 9,18)( 3,12, 6,15, 7,16)$
6C-1 $6^{3}$ $27$ $6$ $15$ $( 1,17, 8,14, 5,10)( 2,18, 9,13, 4,11)( 3,16, 7,15, 6,12)$
6D1 $6^{3}$ $27$ $6$ $15$ $( 1,10, 4,13, 7,16)( 2,12, 6,14, 8,18)( 3,11, 5,15, 9,17)$
6D-1 $6^{3}$ $27$ $6$ $15$ $( 1,18, 9,15, 6,10)( 2,17, 7,13, 5,12)( 3,16, 8,14, 4,11)$
6E $6^{2},2^{3}$ $54$ $6$ $13$ $( 1,13, 2,14, 3,15)( 4,18, 5,16, 6,17)( 7,11)( 8,10)( 9,12)$
6F $6^{2},2^{3}$ $54$ $6$ $13$ $( 1,15)( 2,14)( 3,13)( 4,18, 6,16, 5,17)( 7,10, 9,12, 8,11)$
6G1 $6^{3}$ $54$ $6$ $15$ $( 1,16, 8,15, 4,11)( 2,18, 9,13, 6,10)( 3,17, 7,14, 5,12)$
6G-1 $6^{3}$ $54$ $6$ $15$ $( 1,11, 4,15, 8,16)( 2,10, 6,13, 9,18)( 3,12, 5,14, 7,17)$
6H1 $6^{3}$ $54$ $6$ $15$ $( 1,17, 9,15, 5,12)( 2,18, 7,14, 4,10)( 3,16, 8,13, 6,11)$
6H-1 $6^{3}$ $54$ $6$ $15$ $( 1,12, 5,15, 9,17)( 2,10, 4,14, 7,18)( 3,11, 6,13, 8,16)$
6I1 $6^{2},3^{2}$ $81$ $6$ $14$ $( 1, 5, 7, 3, 4, 9)( 2, 6, 8)(10,15,16,11,13,17)(12,14,18)$
6I-1 $6^{2},3^{2}$ $81$ $6$ $14$ $( 1, 9, 4, 3, 7, 5)( 2, 8, 6)(10,17,13,11,16,15)(12,18,14)$

Malle's constant $a(G)$:     $1/4$

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

36 x 36 character table

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