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Properties

Label 30T39
30T39 1 15 1->15 23 1->23 2 13 2->13 24 2->24 3 14 3->14 22 3->22 4 4->23 27 4->27 5 5->24 25 5->25 6 6->22 26 6->26 7 21 7->21 29 7->29 8 19 8->19 30 8->30 9 20 9->20 28 9->28 10 10->1 10->29 11 11->2 11->30 12 12->3 12->28 13->5 13->27 14->6 14->25 15->4 15->26 16 16->6 16->8 17 17->4 17->9 18 18->5 18->7 19->1 19->11 20->2 20->12 21->3 21->10 22->10 22->15 23->11 23->13 24->12 24->14 25->7 25->16 26->8 26->17 27->9 27->18 28->18 28->21 29->16 29->19 30->17 30->20
Degree $30$
Order $150$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5\times C_{15}$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(30, 39);
 

Group invariants

Abstract group:  $D_5\times C_{15}$
Copy content magma:IdentifyGroup(G);
 
Order:  $150=2 \cdot 3 \cdot 5^{2}$
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 $n$:  $30$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $39$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $15$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,23,13,5,25,16,8,30,20,12,3,22,15,4,27,18,7,29,19,11,2,24,14,6,26,17,9,28,21,10)$, $(1,15,26,8,19)(2,13,27,9,20)(3,14,25,7,21)(4,23,11,30,17)(5,24,12,28,18)(6,22,10,29,16)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$5$:  $C_5$
$6$:  $C_6$
$10$:  $D_{5}$, $C_{10}$
$15$:  $C_{15}$
$30$:  $D_5\times C_3$, $C_{30}$
$50$:  $D_5\times C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 5: None

Degree 6: $C_6$

Degree 10: $D_5\times C_5$

Degree 15: None

Low degree siblings

30T39

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

60 x 60 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed