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Properties

Label 45T164
Degree $45$
Order $1215$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\wr C_5$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 164);
 

Group invariants

Abstract group:  $C_3\wr C_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $1215=3^{5} \cdot 5$
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 $n$:  $45$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $164$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $9$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,21,37,12,30)(2,20,38,11,29)(3,19,39,10,28)(4,24,42,13,32)(5,22,40,14,33)(6,23,41,15,31)(7,27,45,17,34)(8,26,44,18,36)(9,25,43,16,35)$, $(1,25,6,30,7,32,11,35,13,39,18,41,20,44,22)(2,27,5,29,8,31,10,34,15,37,16,40,19,43,24)(3,26,4,28,9,33,12,36,14,38,17,42,21,45,23)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$5$:  $C_5$
$15$:  $C_{15}$
$405$:  15T26

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $C_5$

Degree 9: None

Degree 15: $C_{15}$, 15T26, 15T36 x 2

Low degree siblings

15T36 x 16, 45T164 x 7, 45T165 x 16, 45T166 x 32, 45T167 x 32, 45T168 x 64, 45T169 x 64

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

63 x 63 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed