Properties

Label 36T88079
Degree $36$
Order $459165024$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^8.(C_3^4:C_6.D_6^2)$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(36, 88079);
 
Copy content sage:G = TransitiveGroup(36, 88079)
 
Copy content oscar:G = transitive_group(36, 88079)
 
Copy content gap:G := TransitiveGroup(36, 88079);
 

Group invariants

Abstract group:  $C_3^8.(C_3^4:C_6.D_6^2)$
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:  $459165024=2^{5} \cdot 3^{15}$
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 $t$:  $88079$
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);
 
$\card{\Aut(F/K)}$:  $3$
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:  $(1,29,15,18)(2,28,14,17)(3,30,13,16)(4,26,6,25,5,27)(7,11,32,22,9,12,31,23,8,10,33,24)(19,35,21,36,20,34)$, $(1,7,14,31,2,8,13,32,3,9,15,33)(4,11,30,34)(5,10,29,36)(6,12,28,35)(16,22,17,23,18,24)(19,25)(20,26)(21,27)$, $(1,15,3,13,2,14)(4,30,5,29,6,28)(7,9,8)(10,12,11)(16,18,17)(31,32,33)(34,36,35)$, $(1,19,3,21,2,20)(4,10,16,23)(5,12,18,22)(6,11,17,24)(7,15,33,26,8,14,31,27,9,13,32,25)(28,35)(29,36)(30,34)$
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 15
$4$:  $C_2^2$ x 35
$6$:  $S_3$ x 3
$8$:  $C_2^3$ x 15
$12$:  $D_{6}$ x 21
$16$:  $C_2^4$
$24$:  $S_3 \times C_2^2$ x 21
$32$:  $Q_8:C_2^2$
$36$:  $S_3^2$ x 3
$48$:  24T30 x 3
$72$:  12T37 x 9
$96$:  24T95 x 3
$108$:  12T71
$144$:  24T232 x 3
$216$:  24T548 x 3
$288$:  24T582 x 3
$864$:  24T2612
$2592$:  12T242
$7776$:  24T9718 x 3
$23328$:  24T12660 x 3
$69984$:  24T16091
$5668704$:  36T54854

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: None

Degree 9: None

Degree 12: 12T242

Degree 18: None

Low degree siblings

36T88079 x 26

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