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Properties

Label 36T37
36T37 1 17 1->17 24 1->24 2 18 2->18 23 2->23 3 19 3->19 22 3->22 4 20 4->20 21 4->21 5 14 5->14 27 5->27 6 13 6->13 28 6->28 7 16 7->16 25 7->25 8 15 8->15 26 8->26 9 9->19 33 9->33 10 10->20 34 10->34 11 11->18 35 11->35 12 12->17 36 12->36 13->5 13->12 14->6 14->11 15->7 15->10 16->8 16->9 17->2 17->16 18->1 18->15 19->4 19->14 20->3 20->13 21->8 21->23 22->7 22->24 23->5 23->22 24->6 24->21 32 25->32 25->35 31 26->31 26->36 29 27->29 27->34 30 28->30 28->33 29->3 29->28 30->4 30->27 31->2 31->25 32->1 32->26 33->10 33->31 34->9 34->32 35->12 35->30 36->11 36->29
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_6.D_6$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(36, 37);
 
Copy content sage:G = TransitiveGroup(36, 37)
 
Copy content oscar:G = transitive_group(36, 37)
 
Copy content gap:G := TransitiveGroup(36, 37);
 

Group invariants

Abstract group:  $C_6.D_6$
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 $t$:  $37$
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:  $(1,24,6,28,33,31,2,23,5,27,34,32)(3,22,7,25,35,30,4,21,8,26,36,29)(9,19,14,11,18,15,10,20,13,12,17,16)$, $(1,17,2,18)(3,19,4,20)(5,14,6,13)(7,16,8,15)(9,33,10,34)(11,35,12,36)(21,23,22,24)(25,32,26,31)(27,29,28,30)$
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
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$ x 2
$8$:  $C_4\times C_2$
$12$:  $D_{6}$ x 2, $C_3 : C_4$ x 2
$24$:  $S_3 \times C_4$, 24T6
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 2

Degree 4: $C_4$

Degree 6: $S_3$, $D_{6}$

Degree 9: $S_3^2$

Degree 12: $C_3 : C_4$, $S_3 \times C_4$

Degree 18: $S_3^2$

Low degree siblings

24T60, 36T37

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

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 3A 3B 3C 4A1 4A-1 4B1 4B-1 6A 6B 6C 6D 6E 12A1 12A-1
Size 1 1 3 3 2 2 4 3 3 9 9 2 2 4 6 6 6 6
2 P 1A 1A 1A 1A 3A 3B 3C 2A 2A 2A 2A 3A 3B 3C 3A 3A 6B 6B
3 P 1A 2A 2B 2C 1A 1A 1A 4A-1 4A1 4B-1 4B1 2A 2A 2A 2B 2C 4A1 4A-1
Type
72.20.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.20.1b R 1 1 −1 −1 1 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
72.20.1c R 1 1 −1 −1 1 1 1 1 1 −1 −1 1 1 1 −1 −1 1 1
72.20.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1
72.20.1e1 C 1 −1 −1 1 1 1 1 −i i −i i −1 −1 −1 −1 1 i −i
72.20.1e2 C 1 −1 −1 1 1 1 1 i −i i −i −1 −1 −1 −1 1 −i i
72.20.1f1 C 1 −1 1 −1 1 1 1 −i i i −i −1 −1 −1 1 −1 i −i
72.20.1f2 C 1 −1 1 −1 1 1 1 i −i −i i −1 −1 −1 1 −1 −i i
72.20.2a R 2 2 0 0 2 −1 −1 2 2 0 0 2 −1 −1 0 0 −1 −1
72.20.2b R 2 2 2 2 −1 2 −1 0 0 0 0 −1 2 −1 −1 −1 0 0
72.20.2c R 2 2 −2 −2 −1 2 −1 0 0 0 0 −1 2 −1 1 1 0 0
72.20.2d R 2 2 0 0 2 −1 −1 −2 −2 0 0 2 −1 −1 0 0 1 1
72.20.2e S 2 −2 −2 2 −1 2 −1 0 0 0 0 1 −2 1 1 −1 0 0
72.20.2f S 2 −2 2 −2 −1 2 −1 0 0 0 0 1 −2 1 −1 1 0 0
72.20.2g1 C 2 −2 0 0 2 −1 −1 −2i 2i 0 0 −2 1 1 0 0 −i i
72.20.2g2 C 2 −2 0 0 2 −1 −1 2i −2i 0 0 −2 1 1 0 0 i −i
72.20.4a R 4 4 0 0 −2 −2 1 0 0 0 0 −2 −2 1 0 0 0 0
72.20.4b S 4 −4 0 0 −2 −2 1 0 0 0 0 2 2 −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