↑

Properties

Label 18T113
18T113 1 3 1->3 12 1->12 2 4 2->4 11 2->11 7 3->7 9 3->9 8 4->8 10 4->10 5 5->9 16 5->16 6 6->10 15 6->15 7->5 14 7->14 8->6 13 8->13 9->11 18 9->18 10->12 17 10->17 11->13 11->15 12->14 12->16 13->4 13->18 14->3 14->17 15->2 15->8 16->1 16->7 17->2 17->6 18->1 18->5
Degree $18$
Order $288$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $A_4\wr C_2$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(18, 113);
 
Copy content sage:G = TransitiveGroup(18, 113)
 
Copy content oscar:G = transitive_group(18, 113)
 
Copy content gap:G := TransitiveGroup(18, 113);
 

Group invariants

Abstract group:  $A_4\wr C_2$
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:  $288=2^{5} \cdot 3^{2}$
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 $t$:  $113$
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:  $(1,3,9,11,13,18)(2,4,10,12,14,17)(5,16,7)(6,15,8)$, $(1,12,16)(2,11,15)(3,7,14)(4,8,13)(5,9,18)(6,10,17)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: None

Degree 9: $S_3\times C_3$

Low degree siblings

8T42, 12T126, 12T128, 12T129, 16T708, 18T112, 24T692, 24T694, 24T695, 24T702, 24T703, 24T704, 32T9306, 36T316, 36T318, 36T456, 36T457, 36T458, 36T459

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 $2^{6},1^{6}$ $6$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)(13,14)(15,16)(17,18)$
2B $2^{4},1^{10}$ $9$ $2$ $4$ $( 3, 4)( 5, 6)( 7, 8)(17,18)$
2C $2^{6},1^{6}$ $12$ $2$ $6$ $( 3, 8)( 4, 7)( 5,17)( 6,18)(11,15)(12,16)$
3A1 $3^{6}$ $8$ $3$ $12$ $( 1, 6, 4)( 2, 5, 3)( 7,12, 9)( 8,11,10)(13,16,18)(14,15,17)$
3A-1 $3^{6}$ $8$ $3$ $12$ $( 1, 4, 6)( 2, 3, 5)( 7, 9,12)( 8,10,11)(13,18,16)(14,17,15)$
3B1 $3^{6}$ $16$ $3$ $12$ $( 1, 9,13)( 2,10,14)( 3,12,17)( 4,11,18)( 5, 8,16)( 6, 7,15)$
3B-1 $3^{6}$ $16$ $3$ $12$ $( 1,13, 9)( 2,14,10)( 3,17,12)( 4,18,11)( 5,16, 8)( 6,15, 7)$
3C $3^{6}$ $32$ $3$ $12$ $( 1,12,15)( 2,11,16)( 3, 8,14)( 4, 7,13)( 5, 9,18)( 6,10,17)$
4A $4^{2},2^{4},1^{2}$ $36$ $4$ $10$ $( 3, 7, 4, 8)( 5,17, 6,18)( 9,10)(11,15)(12,16)(13,14)$
6A1 $6^{2},3^{2}$ $24$ $6$ $14$ $( 1, 3, 6, 2, 4, 5)( 7, 9,12)( 8,10,11)(13,17,16,14,18,15)$
6A-1 $6^{2},3^{2}$ $24$ $6$ $14$ $( 1, 5, 4, 2, 6, 3)( 7,12, 9)( 8,11,10)(13,15,18,14,16,17)$
6B1 $6^{2},3^{2}$ $48$ $6$ $14$ $( 1,13, 9)( 2,14,10)( 3, 5,12, 8,17,16)( 4, 6,11, 7,18,15)$
6B-1 $6^{2},3^{2}$ $48$ $6$ $14$ $( 1, 9,13)( 2,10,14)( 3,16,17, 8,12, 5)( 4,15,18, 7,11, 6)$

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

1A 2A 2B 2C 3A1 3A-1 3B1 3B-1 3C 4A 6A1 6A-1 6B1 6B-1
Size 1 6 9 12 8 8 16 16 32 36 24 24 48 48
2 P 1A 1A 1A 1A 3A-1 3A1 3B-1 3B1 3C 2B 3A1 3A-1 3B1 3B-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 4A 2A 2A 2C 2C
Type
288.1025.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
288.1025.1b R 1 1 1 −1 1 1 1 1 1 −1 1 1 −1 −1
288.1025.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1
288.1025.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3
288.1025.1d1 C 1 1 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1
288.1025.1d2 C 1 1 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3
288.1025.2a R 2 2 2 0 −1 −1 2 2 −1 0 −1 −1 0 0
288.1025.2b1 C 2 2 2 0 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 0 −ζ3−1 −ζ3 0 0
288.1025.2b2 C 2 2 2 0 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 0 −ζ3 −ζ3−1 0 0
288.1025.6a R 6 2 −2 0 3 3 0 0 0 0 −1 −1 0 0
288.1025.6b1 C 6 2 −2 0 3ζ3−1 3ζ3 0 0 0 0 −ζ3 −ζ3−1 0 0
288.1025.6b2 C 6 2 −2 0 3ζ3 3ζ3−1 0 0 0 0 −ζ3−1 −ζ3 0 0
288.1025.9a R 9 −3 1 3 0 0 0 0 0 −1 0 0 0 0
288.1025.9b R 9 −3 1 −3 0 0 0 0 0 1 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