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Properties

Label 20T530
20T530 1 10 1->10 15 1->15 2 9 2->9 16 2->16 3 11 3->11 14 3->14 4 12 4->12 13 4->13 5 5->3 6 5->6 6->4 7 7->2 8 8->1 17 9->17 20 9->20 18 10->18 19 10->19 11->18 11->19 12->17 12->20 13->4 13->10 14->3 14->9 15->1 15->11 16->2 16->12 17->8 17->14 18->7 18->13 19->6 19->16 20->5 20->15
Degree $20$
Order $20480$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^9.(C_2\times F_5)$

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Copy content magma:G := TransitiveGroup(20, 530);
 

Group invariants

Abstract group:  $C_2^9.(C_2\times F_5)$
Copy content magma:IdentifyGroup(G);
 
Order:  $20480=2^{12} \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $20$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $530$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,10,18,13,4,12,20,15)(2,9,17,14,3,11,19,16)(5,6)$, $(1,15,11,18,7,2,16,12,17,8)(3,14,9,20,5)(4,13,10,19,6)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$
$80$:  $D_{10}:C_4$
$320$:  $(C_2^4 : C_5):C_4$
$640$:  $((C_2^4 : C_5):C_4)\times C_2$
$1280$:  20T191
$10240$:  $C_2^8.(C_2\times F_5)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $F_5$

Degree 10: $(C_2^4 : C_5):C_4$

Low degree siblings

20T514 x 4, 20T530 x 3, 40T11336 x 4, 40T11339 x 2, 40T11360 x 2, 40T11364 x 2, 40T11373 x 2, 40T11378 x 2, 40T11390 x 4, 40T11391 x 4, 40T12858 x 2, 40T12860 x 4, 40T12876 x 2, 40T12878 x 2, 40T12880 x 2, 40T12988 x 2, 40T12991 x 2, 40T12992 x 4, 40T13368 x 2, 40T13429 x 2, 40T13440 x 4, 40T13454 x 4, 40T13593 x 2, 40T13608 x 2, 40T14049 x 4, 40T14050 x 4, 40T14163 x 2, 40T14167 x 2, 40T14170 x 2, 40T14175 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 magma:ConjugacyClasses(G);
 

Character table

74 x 74 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed