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Properties

Label 45T416
Degree $45$
Order $6750$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_5\times C_{15}^2):C_6$

Related objects

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Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 416);
 

Group invariants

Abstract group:  $(C_5\times C_{15}^2):C_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $6750=2 \cdot 3^{3} \cdot 5^{3}$
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 $t$:  $416$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,33,35,21,4,7,38,23,27,10,41,43,28,14,17)(2,32,34,20,6,8,39,24,26,12,40,45,29,13,16)(3,31,36,19,5,9,37,22,25,11,42,44,30,15,18)$, $(1,41,25,38,4,18)(2,40,27,39,6,17)(3,42,26,37,5,16)(7,12,32,35,29,13)(8,11,31,34,30,15)(9,10,33,36,28,14)(19,22,45)(20,24,43)(21,23,44)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$ x 4
$6$:  $C_6$ x 4
$9$:  $C_3^2$
$10$:  $D_{5}$
$18$:  $C_6 \times C_3$
$27$:  $C_3^2:C_3$
$30$:  $D_5\times C_3$ x 4
$54$:  18T15
$90$:  45T8
$150$:  $(C_5^2 : C_3):C_2$
$270$:  45T35
$450$:  45T65
$750$:  15T30
$1350$:  45T171
$2250$:  45T238

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Degree 9: $C_3^2:C_3$

Degree 15: 15T30

Low degree siblings

45T416 x 23

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