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Properties

Label 45T11
Degree $45$
Order $135$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times \He_3$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 11);
 

Group invariants

Abstract group:  $C_5\times \He_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $135=3^{3} \cdot 5$
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:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $45$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $11$
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:  $(1,34,23,10,45,33,21,9,41,30,17,5,39,25,14)(2,36,22,12,44,32,20,7,40,29,18,4,37,27,13)(3,35,24,11,43,31,19,8,42,28,16,6,38,26,15)$, $(1,45,40,39,34,32,30,25,22,21,17,13,10,9,4)(2,44,42,37,36,31,29,27,24,20,18,15,12,7,6)(3,43,41,38,35,33,28,26,23,19,16,14,11,8,5)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$ x 4
$5$:  $C_5$
$9$:  $C_3^2$
$15$:  $C_{15}$ x 4
$27$:  $C_3^2:C_3$
$45$:  45T2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $C_5$

Degree 9: $C_3^2:C_3$

Degree 15: $C_{15}$

Low degree siblings

45T11 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

55 x 55 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed