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Properties

Label 36T704
36T704 1 3 1->3 19 1->19 2 21 2->21 20 3->20 4 28 4->28 34 4->34 5 30 5->30 35 5->35 6 29 6->29 36 6->36 7 7->29 7->36 8 8->30 8->35 9 9->28 9->34 10 14 10->14 33 10->33 11 13 11->13 32 11->32 12 15 12->15 31 12->31 13->32 14->15 14->31 15->33 16 23 16->23 17 18 17->18 24 17->24 22 18->22 19->5 19->24 20->6 20->23 21->4 21->22 27 22->27 26 23->26 25 24->25 25->7 26->9 26->27 27->8 28->12 29->10 30->11 31->17 32->18 33->16 34->2 35->1 36->3
Degree $36$
Order $432$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $\He_3:\SD_{16}$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(36, 704);
 
Copy content sage:G = TransitiveGroup(36, 704)
 
Copy content oscar:G = transitive_group(36, 704)
 
Copy content gap:G := TransitiveGroup(36, 704);
 

Group invariants

Abstract group:  $\He_3:\SD_{16}$
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:  $432=2^{4} \cdot 3^{3}$
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 $n$:  $36$
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 $t$:  $704$
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(36).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(36), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(36), G));
 
Generators:  $(1,19,5,35)(2,21,4,34)(3,20,6,36)(7,29,10,14,31,17,24,25)(8,30,11,13,32,18,22,27)(9,28,12,15,33,16,23,26)$, $(1,3)(4,28)(5,30)(6,29)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(14,15)(17,18)(19,24)(20,23)(21,22)(26,27)$
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
$4$:  $C_2^2$
$8$:  $D_{4}$
$16$:  $QD_{16}$
$144$:  $(C_3^2:C_8):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: None

Degree 9: None

Degree 12: 12T84

Degree 18: None

Low degree siblings

27T141

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{36}$ $1$ $1$ $0$ $()$
2A $2^{12},1^{12}$ $9$ $2$ $12$ $( 1,26)( 2,27)( 3,25)( 4,17)( 5,18)( 6,16)( 7,19)( 8,20)( 9,21)(10,23)(11,24)(12,22)$
2B $2^{16},1^{4}$ $36$ $2$ $16$ $( 1,14)( 2,13)( 3,15)( 4, 6)( 7,36)( 8,35)( 9,34)(10,21)(11,20)(12,19)(17,18)(22,33)(23,32)(24,31)(26,27)(28,29)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,24,23)(25,27,26)(28,29,30)(31,33,32)(34,36,35)$
3B $3^{11},1^{3}$ $24$ $3$ $22$ $( 1, 3, 2)( 4,29,18)( 5,30,16)( 6,28,17)( 7,33,21)( 8,31,19)( 9,32,20)(10,36,22)(11,34,23)(12,35,24)(13,14,15)$
4A $4^{6},2^{6}$ $18$ $4$ $24$ $( 1,29)( 2,28)( 3,30)( 4,27,16,14)( 5,26,17,13)( 6,25,18,15)( 7,23)( 8,24)( 9,22)(10,31,34,20)(11,32,35,21)(12,33,36,19)$
4B $4^{6},2^{6}$ $36$ $4$ $24$ $( 1,23,26,10)( 2,24,27,11)( 3,22,25,12)( 4,19,17, 7)( 5,21,18, 9)( 6,20,16, 8)(13,36)(14,34)(15,35)(28,33)(29,32)(30,31)$
6A $6^{4},3^{4}$ $18$ $6$ $28$ $( 1,27, 3,26, 2,25)( 4,16, 5,17, 6,18)( 7,20, 9,19, 8,21)(10,24,12,23,11,22)(13,14,15)(28,30,29)(31,32,33)(34,35,36)$
6B $6^{5},3,2,1$ $72$ $6$ $28$ $( 1,13, 3,14, 2,15)( 4,17,29, 6,18,28)( 5,16,30)( 7,10,33,36,21,22)( 8,12,31,35,19,24)( 9,11,32,34,20,23)(26,27)$
8A1 $8^{3},4^{3}$ $54$ $8$ $30$ $( 1, 7,29,23)( 2, 9,28,22)( 3, 8,30,24)( 4,34,27,20,16,10,14,31)( 5,35,26,21,17,11,13,32)( 6,36,25,19,18,12,15,33)$
8A-1 $8^{3},4^{3}$ $54$ $8$ $30$ $( 1,23,29, 7)( 2,22,28, 9)( 3,24,30, 8)( 4,31,14,10,16,20,27,34)( 5,32,13,11,17,21,26,35)( 6,33,15,12,18,19,25,36)$
12A $12^{2},6^{2}$ $36$ $12$ $32$ $( 1,29,26, 4, 2,28,27, 6, 3,30,25, 5)( 7,10,31,34, 8,11,32,35, 9,12,33,36)(13,18,15,16,14,17)(19,22,21,24,20,23)$
12B1 $12^{2},6^{2}$ $36$ $12$ $32$ $( 1,12,27,23, 3,11,26,22, 2,10,25,24)( 4, 9,16,19, 5, 8,17,21, 6, 7,18,20)(13,35,14,36,15,34)(28,32,30,33,29,31)$
12B-1 $12^{2},6^{2}$ $36$ $12$ $32$ $( 1,24,25,10, 2,22,26,11, 3,23,27,12)( 4,20,18, 7, 6,21,17, 8, 5,19,16, 9)(13,34,15,36,14,35)(28,31,29,33,30,32)$

Malle's constant $a(G)$:     $1/12$

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 3A 3B 4A 4B 6A 6B 8A1 8A-1 12A 12B1 12B-1
Size 1 9 36 2 24 18 36 18 72 54 54 36 36 36
2 P 1A 1A 1A 3A 3B 2A 2A 3A 3B 4A 4A 6A 6A 6A
3 P 1A 2A 2B 1A 1A 4A 4B 2A 2B 8A1 8A-1 4A 4B 4B
Type
432.520.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
432.520.1b R 1 1 −1 1 1 1 −1 1 −1 1 1 −1 −1 1
432.520.1c R 1 1 −1 1 1 1 1 1 −1 −1 −1 1 1 1
432.520.1d R 1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1 1
432.520.2a R 2 2 0 2 2 −2 0 2 0 0 0 0 0 −2
432.520.2b1 C 2 −2 0 2 2 0 0 −2 0 −ζ8−ζ83 ζ8+ζ83 0 0 0
432.520.2b2 C 2 −2 0 2 2 0 0 −2 0 ζ8+ζ83 −ζ8−ζ83 0 0 0
432.520.6a R 6 −2 0 −3 0 2 2 1 0 0 0 −1 −1 −1
432.520.6b R 6 −2 0 −3 0 2 −2 1 0 0 0 1 1 −1
432.520.6c1 C 6 −2 0 −3 0 −2 0 1 0 0 0 −1−2ζ3 1+2ζ3 1
432.520.6c2 C 6 −2 0 −3 0 −2 0 1 0 0 0 1+2ζ3 −1−2ζ3 1
432.520.8a R 8 0 2 8 −1 0 0 0 −1 0 0 0 0 0
432.520.8b R 8 0 −2 8 −1 0 0 0 1 0 0 0 0 0
432.520.12a R 12 4 0 −6 0 0 0 −2 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