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Properties

Label 18T410
18T410 1 13 1->13 16 1->16 2 15 2->15 17 2->17 3 14 3->14 18 3->18 4 12 4->12 4->15 5 11 5->11 5->13 6 10 6->10 6->14 7 7->11 7->16 8 8->12 8->17 9 9->10 9->18 10->3 10->5 11->2 11->6 12->1 13->4 13->8 14->9 15->5 15->7 16->1 16->8 17->3 17->7 18->2 18->9
Degree $18$
Order $2916$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $\He_3^2:C_4$

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, 410);
 
Copy content sage:G = TransitiveGroup(18, 410)
 
Copy content oscar:G = transitive_group(18, 410)
 
Copy content gap:G := TransitiveGroup(18, 410);
 

Group invariants

Abstract group:  $\He_3^2:C_4$
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:  $2916=2^{2} \cdot 3^{6}$
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 $t$:  $410$
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:  $(1,16,8,12)(2,17,7,11)(3,18,9,10)(4,15,5,13)(6,14)$, $(1,13,8,17,3,14,9,18,2,15,7,16)(4,12)(5,11,6,10)$
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)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$
$36$:  $C_3^2:C_4$, $C_3\times (C_3 : C_4)$
$108$:  12T72, 12T73
$324$:  12T131

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$

Degree 9: None

Low degree siblings

18T410 x 2, 18T412 x 3, 36T4061 x 3, 36T4063 x 3, 36T4192 x 3, 36T4219 x 3, 36T4260 x 3, 36T4261 x 6

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 $2^{8},1^{2}$ $81$ $2$ $8$ $( 1, 2)( 4, 7)( 5, 8)( 6, 9)(10,16)(11,18)(12,17)(13,14)$
3A $3^{3},1^{9}$ $4$ $3$ $6$ $(10,11,12)(13,15,14)(16,17,18)$
3B $3^{6}$ $4$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,17,18)$
3C1 $3^{2},1^{12}$ $6$ $3$ $4$ $(4,6,5)(7,9,8)$
3C-1 $3^{2},1^{12}$ $6$ $3$ $4$ $(4,5,6)(7,8,9)$
3D1 $3^{4},1^{6}$ $9$ $3$ $8$ $( 4, 5, 6)( 7, 8, 9)(10,12,11)(16,17,18)$
3D-1 $3^{4},1^{6}$ $9$ $3$ $8$ $( 4, 6, 5)( 7, 9, 8)(10,11,12)(16,18,17)$
3E $3^{3},1^{9}$ $12$ $3$ $6$ $(10,13,17)(11,15,18)(12,14,16)$
3F $3^{6}$ $12$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,13,17)(11,15,18)(12,14,16)$
3G $3^{6}$ $12$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,16,15)(11,17,14)(12,18,13)$
3H1 $3^{3},1^{9}$ $12$ $3$ $6$ $(10,13,16)(11,15,17)(12,14,18)$
3H-1 $3^{3},1^{9}$ $12$ $3$ $6$ $(10,13,18)(11,15,16)(12,14,17)$
3I1 $3^{5},1^{3}$ $12$ $3$ $10$ $( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,15,14)(16,17,18)$
3I-1 $3^{5},1^{3}$ $12$ $3$ $10$ $( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,15,14)(16,17,18)$
3J1 $3^{6}$ $12$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,13,16)(11,15,17)(12,14,18)$
3J-1 $3^{6}$ $12$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,13,18)(11,15,16)(12,14,17)$
3K1 $3^{6}$ $12$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,16,13)(11,17,15)(12,18,14)$
3K-1 $3^{6}$ $12$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,16,14)(11,17,13)(12,18,15)$
3L $3^{4},1^{6}$ $18$ $3$ $8$ $( 4, 5, 6)( 7, 8, 9)(13,14,15)(16,17,18)$
3M $3^{6}$ $36$ $3$ $12$ $( 1, 4, 9)( 2, 6, 7)( 3, 5, 8)(10,13,17)(11,15,18)(12,14,16)$
3N1 $3^{5},1^{3}$ $36$ $3$ $10$ $( 4, 5, 6)( 7, 8, 9)(10,13,16)(11,15,17)(12,14,18)$
3N-1 $3^{5},1^{3}$ $36$ $3$ $10$ $( 4, 6, 5)( 7, 9, 8)(10,13,18)(11,15,16)(12,14,17)$
3O1 $3^{5},1^{3}$ $36$ $3$ $10$ $( 4, 5, 6)( 7, 8, 9)(10,13,17)(11,15,18)(12,14,16)$
3O-1 $3^{5},1^{3}$ $36$ $3$ $10$ $( 4, 6, 5)( 7, 9, 8)(10,13,17)(11,15,18)(12,14,16)$
3P1 $3^{5},1^{3}$ $36$ $3$ $10$ $( 4, 5, 6)( 7, 8, 9)(10,13,18)(11,15,16)(12,14,17)$
3P-1 $3^{5},1^{3}$ $36$ $3$ $10$ $( 4, 6, 5)( 7, 9, 8)(10,13,16)(11,15,17)(12,14,18)$
3Q1 $3^{6}$ $36$ $3$ $12$ $( 1, 4, 7)( 2, 6, 8)( 3, 5, 9)(10,13,16)(11,15,17)(12,14,18)$
3Q-1 $3^{6}$ $36$ $3$ $12$ $( 1, 4, 8)( 2, 6, 9)( 3, 5, 7)(10,13,18)(11,15,16)(12,14,17)$
3R1 $3^{6}$ $36$ $3$ $12$ $( 1, 4, 7)( 2, 6, 8)( 3, 5, 9)(10,13,17)(11,15,18)(12,14,16)$
3R-1 $3^{6}$ $36$ $3$ $12$ $( 1, 4, 8)( 2, 6, 9)( 3, 5, 7)(10,13,17)(11,15,18)(12,14,16)$
3S1 $3^{6}$ $36$ $3$ $12$ $( 1, 4, 7)( 2, 6, 8)( 3, 5, 9)(10,13,18)(11,15,16)(12,14,17)$
3S-1 $3^{6}$ $36$ $3$ $12$ $( 1, 4, 7)( 2, 6, 8)( 3, 5, 9)(10,16,13)(11,17,15)(12,18,14)$
3T1 $3^{6}$ $36$ $3$ $12$ $( 1, 4, 7)( 2, 6, 8)( 3, 5, 9)(10,16,15)(11,17,14)(12,18,13)$
3T-1 $3^{6}$ $36$ $3$ $12$ $( 1, 4, 8)( 2, 6, 9)( 3, 5, 7)(10,16,15)(11,17,14)(12,18,13)$
4A1 $4^{4},2$ $243$ $4$ $13$ $( 1,15, 3,13)( 2,14)( 4,11, 8,17)( 5,10, 9,18)( 6,12, 7,16)$
4A-1 $4^{4},2$ $243$ $4$ $13$ $( 1,13, 3,15)( 2,14)( 4,17, 8,11)( 5,18, 9,10)( 6,16, 7,12)$
6A1 $6^{2},2^{2},1^{2}$ $81$ $6$ $12$ $( 1, 3)( 4, 7, 5, 8, 6, 9)(10,17,12,18,11,16)(13,15)$
6A-1 $6^{2},2^{2},1^{2}$ $81$ $6$ $12$ $( 1, 3)( 4, 9, 6, 8, 5, 7)(10,16,11,18,12,17)(13,15)$
6B $6^{2},2^{2},1^{2}$ $162$ $6$ $12$ $( 2, 3)( 4, 8, 5, 9, 6, 7)(11,12)(13,16,14,17,15,18)$
6C1 $6,2^{5},1^{2}$ $162$ $6$ $10$ $( 1, 2)( 4, 8, 6, 7, 5, 9)(10,16)(11,18)(12,17)(13,14)$
6C-1 $6,2^{5},1^{2}$ $162$ $6$ $10$ $( 1, 2)( 4, 9, 5, 7, 6, 8)(10,16)(11,18)(12,17)(13,14)$
12A1 $12,4,2$ $243$ $12$ $15$ $( 1,13, 3,15)( 2,14)( 4,18, 7,11, 5,16, 8,10, 6,17, 9,12)$
12A-1 $12,4,2$ $243$ $12$ $15$ $( 1,15, 3,13)( 2,14)( 4,12, 9,17, 6,10, 8,16, 5,11, 7,18)$
12A5 $12,4,2$ $243$ $12$ $15$ $( 1,13, 3,15)( 2,14)( 4,16, 9,11, 6,18, 8,12, 5,17, 7,10)$
12A-5 $12,4,2$ $243$ $12$ $15$ $( 1,15, 3,13)( 2,14)( 4,10, 7,17, 5,12, 8,18, 6,11, 9,16)$

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

46 x 46 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