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Properties

Label 20T261
20T261 1 8 1->8 11 1->11 18 1->18 2 7 2->7 12 2->12 17 2->17 3 5 3->5 9 3->9 19 3->19 4 6 4->6 10 4->10 20 4->20 5->1 5->1 5->10 6->2 6->2 6->9 7->3 7->4 7->12 8->3 8->4 8->11 9->10 9->18 9->19 10->17 10->20 11->12 11->18 11->20 12->17 12->19 13 13->1 13->8 14 13->14 14->2 14->7 15 15->4 15->6 16 16->3 16->5 17->5 17->14 17->15 18->6 18->13 18->16 19->8 19->13 19->16 20->7 20->14 20->15
Degree $20$
Order $2560$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4\times C_2^4:F_5$

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Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 261);
 
Copy content sage:G = TransitiveGroup(20, 261)
 
Copy content oscar:G = transitive_group(20, 261)
 

Group invariants

Abstract group:  $D_4\times C_2^4:F_5$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $2560=2^{9} \cdot 5$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar: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)
 
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)
 
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)
 
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
 

Group action invariants

Degree $n$:  $20$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Transitive number $t$:  $261$
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]
 
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)
 
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)
 
$\card{\Aut(F/K)}$:  $2$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(20).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(20), G)[1])
 
Generators:  $(1,8,11,20,15,4,6,9,18,13)(2,7,12,19,16,3,5,10,17,14)$, $(1,18,16,5)(2,17,15,6)(3,19,13,8)(4,20,14,7)(9,10)(11,12)$, $(1,11,18,6,2,12,17,5)(3,9,19,8,4,10,20,7)(13,14)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$, $Q_8:C_2$, $C_4\times C_2^2$
$20$:  $F_5$
$32$:  $C_4 \times D_4$
$40$:  $F_{5}\times C_2$ x 3
$80$:  20T16
$160$:  20T42
$320$:  $(C_2^4 : C_5):C_4$
$640$:  $((C_2^4 : C_5):C_4)\times C_2$ x 3
$1280$:  20T196

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $F_5$

Degree 10: $F_{5}\times C_2$

Low degree siblings

20T261 x 3, 40T1834 x 2, 40T1835 x 2, 40T1840 x 2, 40T1841 x 2, 40T1844, 40T1849 x 2, 40T1850 x 2, 40T2177 x 4, 40T2178 x 4, 40T2179 x 4, 40T2180 x 4, 40T2249 x 2, 40T2258 x 2, 40T2282 x 2, 40T2283 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)
 

Character table

55 x 55 character table

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 

Regular extensions

Data not computed