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Properties

Label 30T104
30T104 1 10 1->10 25 1->25 2 11 2->11 27 2->27 3 12 3->12 26 3->26 4 13 4->13 30 4->30 5 14 5->14 29 5->29 6 15 6->15 28 6->28 7 7->2 18 7->18 8 8->3 17 8->17 9 9->1 16 9->16 10->5 21 10->21 11->6 20 11->20 12->4 19 12->19 13->9 24 13->24 14->7 23 14->23 15->8 22 15->22 16->11 16->24 17->10 17->22 18->12 18->23 19->14 19->26 20->13 20->27 21->15 21->25 22->1 22->28 23->3 23->29 24->2 24->30 25->6 25->16 26->5 26->17 27->4 27->18 28->9 28->20 29->8 29->21 30->7 30->19
Degree $30$
Order $450$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{15}\times D_{15}$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(30, 104);
 

Group invariants

Abstract group:  $C_{15}\times D_{15}$
Copy content magma:IdentifyGroup(G);
 
Order:  $450=2 \cdot 3^{2} \cdot 5^{2}$
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 $t$:  $104$
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:  $(1,25,6,28,9,16,11,20,13,24,2,27,4,30,7,18,12,19,14,23,3,26,5,29,8,17,10,21,15,22)$, $(1,10,5,14,7,2,11,6,15,8,3,12,4,13,9)(16,24,30,19,26,17,22,28,20,27,18,23,29,21,25)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$5$:  $C_5$
$6$:  $S_3$, $C_6$
$10$:  $D_{5}$, $C_{10}$
$15$:  $C_{15}$
$18$:  $S_3\times C_3$
$30$:  $D_{15}$, $D_5\times C_3$, $S_3 \times C_5$, $C_{30}$
$50$:  $D_5\times C_5$
$90$:  30T15, 30T16
$150$:  30T36, 30T39

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: None

Degree 6: $S_3\times C_3$

Degree 10: $D_5\times C_5$

Degree 15: None

Low degree siblings

30T104 x 3

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

135 x 135 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed