↑

Properties

Label 24T92
24T92 1 4 1->4 9 1->9 17 1->17 2 3 2->3 10 2->10 18 2->18 6 3->6 12 3->12 13 3->13 5 4->5 11 4->11 14 4->14 5->1 8 5->8 16 5->16 6->2 7 6->7 15 6->15 7->9 24 7->24 8->10 23 8->23 9->12 20 9->20 10->11 19 10->19 11->7 21 11->21 12->8 22 12->22 13->14 13->15 13->21 14->16 14->22 15->16 15->18 15->23 16->17 16->24 17->14 17->18 17->19 18->13 18->20 19->20 19->21 20->22 21->22 21->24 22->23 23->19 23->24 24->20
Degree $24$
Order $96$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_6.C_2^4$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(24, 92);
 
Copy content sage:G = TransitiveGroup(24, 92)
 
Copy content oscar:G = transitive_group(24, 92)
 
Copy content gap:G := TransitiveGroup(24, 92);
 

Group invariants

Abstract group:  $C_6.C_2^4$
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:  $96=2^{5} \cdot 3$
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:  $2$
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 $n$:  $24$
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 $t$:  $92$
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);
 
$\card{\Aut(F/K)}$:  $6$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(24).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(24), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(24), G));
 
Generators:  $(1,9)(2,10)(3,12)(4,11)(5,8)(6,7)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)$, $(1,17)(2,18)(3,13)(4,14)(5,16)(6,15)(7,24)(8,23)(9,20)(10,19)(11,21)(12,22)$, $(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$, $(1,4,5)(2,3,6)(7,9,12,8,10,11)(13,15,18)(14,16,17)(19,21,24,20,22,23)$
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
$3$:  $C_3$
$4$:  $C_2^2$ x 35
$6$:  $C_6$ x 15
$8$:  $C_2^3$ x 15
$12$:  $C_6\times C_2$ x 35
$16$:  $C_2^4$
$24$:  24T3 x 15
$32$:  $Q_8:C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3

Degree 8: $Q_8:C_2^2$

Degree 12: $C_6\times C_2$

Low degree siblings

24T92 x 5

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

51 x 51 character table

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