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Properties

Label 36T30
36T30 1 16 1->16 17 1->17 2 15 2->15 18 2->18 3 14 3->14 20 3->20 4 13 4->13 19 4->19 5 5->18 23 5->23 6 6->17 24 6->24 7 7->19 22 7->22 8 8->20 21 8->21 9 9->13 9->21 10 10->14 10->22 11 11->16 11->23 12 12->15 12->24 31 13->31 35 13->35 32 14->32 36 14->36 29 15->29 33 15->33 30 16->30 34 16->34 27 17->27 17->35 28 18->28 18->36 25 19->25 19->34 26 20->26 20->33 21->28 21->30 22->27 22->29 23->26 23->31 24->25 24->32 25->6 25->9 26->5 26->10 27->7 27->11 28->8 28->12 29->4 29->12 30->3 30->11 31->2 31->9 32->1 32->10 33->3 33->6 34->4 34->5 35->1 35->8 36->2 36->7
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2^2:C_{18}$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(36, 30);
 
Copy content sage:G = TransitiveGroup(36, 30)
 
Copy content oscar:G = transitive_group(36, 30)
 
Copy content gap:G := TransitiveGroup(36, 30);
 

Group invariants

Abstract group:  $C_2^2:C_{18}$
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 $n$:  $36$
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 $t$:  $30$
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);
 
$\card{\Aut(F/K)}$:  $12$
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:  $(1,17,27,11,16,34,5,23,31,2,18,28,12,15,33,6,24,32)(3,20,26,10,14,36,7,22,29,4,19,25,9,13,35,8,21,30)$, $(1,16,30,11,23,26,5,18,36,2,15,29,12,24,25,6,17,35)(3,14,32,10,22,27,7,19,34,4,13,31,9,21,28,8,20,33)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$9$:  $C_9$
$12$:  $A_4$
$18$:  $C_{18}$
$24$:  $A_4\times C_2$
$36$:  $C_2^2 : C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$, $A_4\times C_2$

Degree 9: $C_9$

Degree 12: $A_4\times C_2$

Degree 18: $C_2^2 : C_9$, 18T26

Low degree siblings

18T26, 36T16

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 $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B $2^{18}$ $3$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)(33,36)(34,35)$
2C $2^{12},1^{12}$ $3$ $2$ $12$ $(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)(33,35)(34,36)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1,12, 5)( 2,11, 6)( 3, 9, 7)( 4,10, 8)(13,22,20)(14,21,19)(15,23,17)(16,24,18)(25,36,30)(26,35,29)(27,33,31)(28,34,32)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1, 5,12)( 2, 6,11)( 3, 7, 9)( 4, 8,10)(13,20,22)(14,19,21)(15,17,23)(16,18,24)(25,30,36)(26,29,35)(27,31,33)(28,32,34)$
6A1 $6^{6}$ $1$ $6$ $30$ $( 1,11, 5, 2,12, 6)( 3,10, 7, 4, 9, 8)(13,21,20,14,22,19)(15,24,17,16,23,18)(25,35,30,26,36,29)(27,34,31,28,33,32)$
6A-1 $6^{6}$ $1$ $6$ $30$ $( 1, 6,12, 2, 5,11)( 3, 8, 9, 4, 7,10)(13,19,22,14,20,21)(15,18,23,16,17,24)(25,29,36,26,30,35)(27,32,33,28,31,34)$
6B1 $6^{6}$ $3$ $6$ $30$ $( 1, 6,12, 2, 5,11)( 3, 8, 9, 4, 7,10)(13,17,22,15,20,23)(14,18,21,16,19,24)(25,31,36,27,30,33)(26,32,35,28,29,34)$
6B-1 $6^{6}$ $3$ $6$ $30$ $( 1,11, 5, 2,12, 6)( 3,10, 7, 4, 9, 8)(13,23,20,15,22,17)(14,24,19,16,21,18)(25,33,30,27,36,31)(26,34,29,28,35,32)$
6C1 $6^{4},3^{4}$ $3$ $6$ $28$ $( 1,12, 5)( 2,11, 6)( 3, 9, 7)( 4,10, 8)(13,24,20,16,22,18)(14,23,19,15,21,17)(25,34,30,28,36,32)(26,33,29,27,35,31)$
6C-1 $6^{4},3^{4}$ $3$ $6$ $28$ $( 1, 5,12)( 2, 6,11)( 3, 7, 9)( 4, 8,10)(13,18,22,16,20,24)(14,17,21,15,19,23)(25,32,36,28,30,34)(26,31,35,27,29,33)$
9A1 $9^{4}$ $4$ $9$ $32$ $( 1,27,16, 5,31,18,12,33,24)( 2,28,15, 6,32,17,11,34,23)( 3,26,14, 7,29,19, 9,35,21)( 4,25,13, 8,30,20,10,36,22)$
9A-1 $9^{4}$ $4$ $9$ $32$ $( 1,24,33,12,18,31, 5,16,27)( 2,23,34,11,17,32, 6,15,28)( 3,21,35, 9,19,29, 7,14,26)( 4,22,36,10,20,30, 8,13,25)$
9A2 $9^{4}$ $4$ $9$ $32$ $( 1,16,31,12,24,27, 5,18,33)( 2,15,32,11,23,28, 6,17,34)( 3,14,29, 9,21,26, 7,19,35)( 4,13,30,10,22,25, 8,20,36)$
9A-2 $9^{4}$ $4$ $9$ $32$ $( 1,33,18, 5,27,24,12,31,16)( 2,34,17, 6,28,23,11,32,15)( 3,35,19, 7,26,21, 9,29,14)( 4,36,20, 8,25,22,10,30,13)$
9A4 $9^{4}$ $4$ $9$ $32$ $( 1,31,24, 5,33,16,12,27,18)( 2,32,23, 6,34,15,11,28,17)( 3,29,21, 7,35,14, 9,26,19)( 4,30,22, 8,36,13,10,25,20)$
9A-4 $9^{4}$ $4$ $9$ $32$ $( 1,18,27,12,16,33, 5,24,31)( 2,17,28,11,15,34, 6,23,32)( 3,19,26, 9,14,35, 7,21,29)( 4,20,25,10,13,36, 8,22,30)$
18A1 $18^{2}$ $4$ $18$ $34$ $( 1,17,27,11,16,34, 5,23,31, 2,18,28,12,15,33, 6,24,32)( 3,20,26,10,14,36, 7,22,29, 4,19,25, 9,13,35, 8,21,30)$
18A-1 $18^{2}$ $4$ $18$ $34$ $( 1,32,24, 6,33,15,12,28,18, 2,31,23, 5,34,16,11,27,17)( 3,30,21, 8,35,13, 9,25,19, 4,29,22, 7,36,14,10,26,20)$
18A5 $18^{2}$ $4$ $18$ $34$ $( 1,34,18, 6,27,23,12,32,16, 2,33,17, 5,28,24,11,31,15)( 3,36,19, 8,26,22, 9,30,14, 4,35,20, 7,25,21,10,29,13)$
18A-5 $18^{2}$ $4$ $18$ $34$ $( 1,15,31,11,24,28, 5,17,33, 2,16,32,12,23,27, 6,18,34)( 3,13,29,10,21,25, 7,20,35, 4,14,30, 9,22,26, 8,19,36)$
18A7 $18^{2}$ $4$ $18$ $34$ $( 1,23,33,11,18,32, 5,15,27, 2,24,34,12,17,31, 6,16,28)( 3,22,35,10,19,30, 7,13,26, 4,21,36, 9,20,29, 8,14,25)$
18A-7 $18^{2}$ $4$ $18$ $34$ $( 1,28,16, 6,31,17,12,34,24, 2,27,15, 5,32,18,11,33,23)( 3,25,14, 8,29,20, 9,36,21, 4,26,13, 7,30,19,10,35,22)$

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 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 18A1 18A-1 18A5 18A-5 18A7 18A-7
Size 1 1 3 3 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 3A-1 3A1 3A-1 3A1 3A1 3A-1 3A-1 3A1 9A2 9A-2 9A4 9A-4 9A-1 9A1 9A1 9A-1 9A-4 9A4 9A-2 9A2
3 P 1A 2A 2B 2C 1A 1A 2A 2A 2B 2B 2C 2C 3A-1 3A1 3A1 3A-1 3A-1 3A1 6A1 6A-1 6A-1 6A1 6A1 6A-1
Type
72.16.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.16.1b R 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
72.16.1c1 C 1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
72.16.1c2 C 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
72.16.1d1 C 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
72.16.1d2 C 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
72.16.1e1 C 1 1 1 1 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ94 ζ92 ζ9−1 ζ9 ζ9−4 ζ9−2 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2
72.16.1e2 C 1 1 1 1 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ9−4 ζ9−2 ζ9 ζ9−1 ζ94 ζ92 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92
72.16.1e3 C 1 1 1 1 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−2 ζ9−1 ζ9−4 ζ94 ζ92 ζ9 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9
72.16.1e4 C 1 1 1 1 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ92 ζ9 ζ94 ζ9−4 ζ9−2 ζ9−1 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1
72.16.1e5 C 1 1 1 1 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9 ζ9−4 ζ92 ζ9−2 ζ9−1 ζ94 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94
72.16.1e6 C 1 1 1 1 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−3 ζ9−1 ζ94 ζ9−2 ζ92 ζ9 ζ9−4 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4
72.16.1f1 C 1 −1 1 −1 ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 ζ94 ζ92 ζ9−1 ζ9 ζ9−4 ζ9−2 −ζ9−4 −ζ94 −ζ9 −ζ9−1 −ζ92 −ζ9−2
72.16.1f2 C 1 −1 1 −1 ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 ζ9−4 ζ9−2 ζ9 ζ9−1 ζ94 ζ92 −ζ94 −ζ9−4 −ζ9−1 −ζ9 −ζ9−2 −ζ92
72.16.1f3 C 1 −1 1 −1 ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 ζ9−2 ζ9−1 ζ9−4 ζ94 ζ92 ζ9 −ζ92 −ζ9−2 −ζ94 −ζ9−4 −ζ9−1 −ζ9
72.16.1f4 C 1 −1 1 −1 ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 ζ92 ζ9 ζ94 ζ9−4 ζ9−2 ζ9−1 −ζ9−2 −ζ92 −ζ9−4 −ζ94 −ζ9 −ζ9−1
72.16.1f5 C 1 −1 1 −1 ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 ζ9 ζ9−4 ζ92 ζ9−2 ζ9−1 ζ94 −ζ9−1 −ζ9 −ζ9−2 −ζ92 −ζ9−4 −ζ94
72.16.1f6 C 1 −1 1 −1 ζ93 ζ9−3 −ζ93 −ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−3 ζ9−1 ζ94 ζ9−2 ζ92 ζ9 ζ9−4 −ζ9 −ζ9−1 −ζ92 −ζ9−2 −ζ94 −ζ9−4
72.16.3a R 3 3 −1 −1 3 3 3 3 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3b R 3 −3 −1 1 3 3 −3 −3 −1 1 1 −1 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3c1 C 3 3 −1 −1 3ζ3−1 3ζ3 3ζ3−1 3ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3c2 C 3 3 −1 −1 3ζ3 3ζ3−1 3ζ3 3ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3d1 C 3 −3 −1 1 3ζ3−1 3ζ3 −3ζ3−1 −3ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3d2 C 3 −3 −1 1 3ζ3 3ζ3−1 −3ζ3 −3ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 0 0 0 0 0 0 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