↑

Properties

Label 36T47667
Degree $36$
Order $2834352$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^7.(C_3^2\wr C_2.D_4)$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(36, 47667);
 
Copy content sage:G = TransitiveGroup(36, 47667)
 
Copy content oscar:G = transitive_group(36, 47667)
 
Copy content gap:G := TransitiveGroup(36, 47667);
 

Group invariants

Abstract group:  $C_3^7.(C_3^2\wr C_2.D_4)$
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:  $2834352=2^{4} \cdot 3^{11}$
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 $t$:  $47667$
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:  $(1,17,21,12,13,28,31,24,25,6,7,35,2,16,20,10,15,30,32,23,26,4,9,36,3,18,19,11,14,29,33,22,27,5,8,34)$, $(1,21,13,8,26,33,3,20,14,7,25,31,2,19,15,9,27,32)(4,36,17,10,29,23,5,34,16,11,28,22,6,35,18,12,30,24)$
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, $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$12$:  $D_{6}$, $C_3 : C_4$ x 2
$16$:  $C_2^2:C_4$
$24$:  $(C_6\times C_2):C_2$ x 2, 24T6
$48$:  24T44
$72$:  $C_3^2:D_4$ x 2
$144$:  12T79 x 2
$216$:  12T116 x 2
$432$:  24T1293 x 2
$1296$:  12T211
$3888$:  24T7301
$11664$:  18T575
$34992$:  36T14150

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Degree 9: None

Degree 12: 12T211

Degree 18: None

Low degree siblings

36T47667 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Conjugacy classes not computed

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

Character table not computed

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