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Properties

Label 30T4733
30T4733 1 8 1->8 17 1->17 2 2->1 15 2->15 3 9 3->9 18 3->18 4 6 4->6 13 4->13 5 5->4 11 5->11 6->3 14 6->14 7 10 7->10 12 7->12 8->5 16 8->16 9->2 19 9->19 20 10->20 11->7 30 11->30 12->8 25 12->25 13->5 21 13->21 27 14->27 15->9 29 15->29 16->4 28 16->28 17->10 22 17->22 23 18->23 19->1 24 19->24 20->2 26 20->26 21->17 22->12 22->29 23->14 23->30 24->13 24->26 25->19 26->22 27->11 27->28 28->18 28->23 29->24 30->16 30->27
Degree $30$
Order $559872000$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $A_6^3.D_6$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(30, 4733);
 

Group invariants

Abstract group:  $A_6^3.D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $559872000=2^{11} \cdot 3^{7} \cdot 5^{3}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $30$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4733$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,17,10,20,2,15,9,19)(3,18)(4,13,5,11,7,12,8,16)(6,14)(22,29,24,26)(23,30,27,28)$, $(1,8,5,4,6,3,9,2)(7,10)(11,30,16,28,18,23,14,27)(12,25,19,24,13,21,17,22)(15,29)(20,26)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 5: None

Degree 6: None

Degree 10: None

Degree 15: None

Low degree siblings

36T88326

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

112 x 112 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed