↑

Properties

Label 36T34
36T34 1 3 1->3 8 1->8 21 1->21 2 4 2->4 7 2->7 22 2->22 6 3->6 24 3->24 5 4->5 23 4->23 5->7 32 5->32 35 5->35 6->8 31 6->31 36 6->36 30 7->30 34 7->34 29 8->29 33 8->33 9 15 9->15 16 9->16 9->21 10 10->15 10->16 10->22 11 13 11->13 14 11->14 11->23 12 12->13 12->14 12->24 19 13->19 26 13->26 20 14->20 25 14->25 18 15->18 28 15->28 17 16->17 27 16->27 17->9 17->29 18->10 18->30 19->12 19->31 20->11 20->32 21->28 22->27 23->26 24->25 25->32 25->35 26->31 26->36 27->29 27->33 28->30 28->34 29->21 30->22 31->24 32->23 33->2 33->36 34->1 34->35 35->3 36->4
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $S_3\times D_6$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(36, 34);
 
Copy content sage:G = TransitiveGroup(36, 34)
 
Copy content oscar:G = transitive_group(36, 34)
 
Copy content gap:G := TransitiveGroup(36, 34);
 

Group invariants

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$ x 2

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 6, $S_3^2$

Degree 9: $S_3^2$

Degree 12: $S_3 \times C_2^2$ x 2, 12T37

Degree 18: $S_3^2$, 18T29 x 2

Low degree siblings

12T37 x 2, 18T29 x 4, 24T73, 36T34, 36T40 x 4

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

Malle's constant $a(G)$:     $1/16$

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 2D 2E 2F 2G 3A 3B 3C 6A 6B 6C 6D 6E 6F 6G
Size 1 1 3 3 3 3 9 9 2 2 4 2 2 4 6 6 6 6
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3B 3C 3A 3B 3C 3B 3B 3A 3A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 1A 2A 2A 2A 2B 2C 2D 2E
Type
72.46.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1b R 1 −1 −1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 −1 1
72.46.1c R 1 −1 −1 1 1 −1 1 −1 1 1 1 −1 −1 −1 −1 1 1 −1
72.46.1d R 1 −1 1 −1 −1 1 1 −1 1 1 1 −1 −1 −1 1 −1 −1 1
72.46.1e R 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1 −1 1 −1
72.46.1f R 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
72.46.1g R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 1 1
72.46.1h R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1
72.46.2a R 2 2 0 0 2 2 0 0 −1 2 −1 −1 2 −1 0 0 −1 −1
72.46.2b R 2 2 2 2 0 0 0 0 2 −1 −1 2 −1 −1 −1 −1 0 0
72.46.2c R 2 −2 −2 2 0 0 0 0 2 −1 −1 −2 1 1 1 −1 0 0
72.46.2d R 2 −2 0 0 −2 2 0 0 −1 2 −1 1 −2 1 0 0 1 −1
72.46.2e R 2 −2 0 0 2 −2 0 0 −1 2 −1 1 −2 1 0 0 −1 1
72.46.2f R 2 −2 2 −2 0 0 0 0 2 −1 −1 −2 1 1 −1 1 0 0
72.46.2g R 2 2 −2 −2 0 0 0 0 2 −1 −1 2 −1 −1 1 1 0 0
72.46.2h R 2 2 0 0 −2 −2 0 0 −1 2 −1 −1 2 −1 0 0 1 1
72.46.4a R 4 4 0 0 0 0 0 0 −2 −2 1 −2 −2 1 0 0 0 0
72.46.4b R 4 −4 0 0 0 0 0 0 −2 −2 1 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