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Properties

Label 18T32
18T32 1 5 1->5 12 1->12 2 6 2->6 11 2->11 3 7 3->7 4 8 4->8 9 5->9 10 6->10 7->10 16 7->16 8->9 15 8->15 17 9->17 18 10->18 11->12 13 11->13 14 12->14 13->2 13->15 14->1 14->16 15->4 16->3 17->5 18->6
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $S_3\times A_4$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(18, 32);
 
Copy content sage:G = TransitiveGroup(18, 32)
 
Copy content oscar:G = transitive_group(18, 32)
 
Copy content gap:G := TransitiveGroup(18, 32);
 

Group invariants

Abstract group:  $S_3\times A_4$
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 $t$:  $32$
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);
 
$\card{\Aut(F/K)}$:  $2$
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:  $(1,12,14)(2,11,13)(3,7,16)(4,8,15)(5,9,17)(6,10,18)$, $(1,5)(2,6)(7,10)(8,9)(11,12)(13,15)(14,16)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$12$:  $A_4$
$18$:  $S_3\times C_3$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: $A_4\times C_2$

Degree 9: $S_3\times C_3$

Low degree siblings

12T43, 18T31, 24T78, 24T83, 36T21, 36T50, 36T51

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

Malle's constant $a(G)$:     $1/6$

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 3A 3B1 3B-1 3C1 3C-1 6A 6B1 6B-1
Size 1 3 3 9 2 4 4 8 8 6 12 12
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3C-1 3C1 3A 3B1 3B-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2A 2B 2B
Type
72.44.1a R 1 1 1 1 1 1 1 1 1 1 1 1
72.44.1b R 1 1 −1 −1 1 1 1 1 1 1 −1 −1
72.44.1c1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 ζ3 ζ3−1
72.44.1c2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 ζ3−1 ζ3
72.44.1d1 C 1 1 −1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 −ζ3 −ζ3−1
72.44.1d2 C 1 1 −1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 −ζ3−1 −ζ3
72.44.2a R 2 2 0 0 −1 2 2 −1 −1 −1 0 0
72.44.2b1 C 2 2 0 0 −1 2ζ3−1 2ζ3 −ζ3 −ζ3−1 −1 0 0
72.44.2b2 C 2 2 0 0 −1 2ζ3 2ζ3−1 −ζ3−1 −ζ3 −1 0 0
72.44.3a R 3 −1 3 −1 3 0 0 0 0 −1 0 0
72.44.3b R 3 −1 −3 1 3 0 0 0 0 −1 0 0
72.44.6a R 6 −2 0 0 −3 0 0 0 0 1 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