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Properties

Label 45T319
Degree $45$
Order $3645$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^5:C_{15}$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 319);
 

Group invariants

Abstract group:  $C_3^5:C_{15}$
Copy content magma:IdentifyGroup(G);
 
Order:  $3645=3^{6} \cdot 5$
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 $t$:  $319$
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:  $(1,10,21,28,39,2,12,20,30,37,3,11,19,29,38)(4,14,22,31,42,6,13,24,33,41,5,15,23,32,40)(7,16,26,35,44,8,17,25,34,43,9,18,27,36,45)$, $(1,34,23,11,43,33,20,8,42,30,16,4,39,27,14)(2,36,22,10,45,32,19,9,41,29,17,6,37,26,13)(3,35,24,12,44,31,21,7,40,28,18,5,38,25,15)$
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
$45$:  45T2
$405$:  15T26
$1215$:  15T36 x 4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $C_5$

Degree 9: None

Degree 15: $C_{15}$, 15T36 x 3

Low degree siblings

45T319 x 63, 45T320 x 96, 45T321 x 768, 45T322 x 384

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

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed