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Properties

Label 45T3793
Degree $45$
Order $105468750$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^7.C_{15}^2:S_3$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 3793);
 

Group invariants

Abstract group:  $C_5^7.C_{15}^2:S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $105468750=2 \cdot 3^{3} \cdot 5^{9}$
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 $t$:  $3793$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $5$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,34,6,27,41,38,5,33,10,26,45,37,4,32,9,30,44,36,3,31,8,29,43,40,2,35,7,28,42,39)(11,17,12,18,13,19,14,20,15,16)(21,23,25,22,24)$, $(1,33,23,5,32,22,4,31,21,3,35,25,2,34,24)(6,36,12,8,38,14,10,40,11,7,37,13,9,39,15)(16,45,29,18,42,26,20,44,28,17,41,30,19,43,27)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$5$:  $C_5$
$6$:  $S_3$, $C_6$
$10$:  $C_{10}$
$15$:  $C_{15}$
$18$:  $S_3\times C_3$
$30$:  $S_3 \times C_5$, $C_{30}$
$54$:  $C_3^2 : C_6$
$90$:  30T15
$150$:  $(C_5^2 : C_3):C_2$
$270$:  45T30
$450$:  45T70
$750$:  15T32
$1350$:  45T174
$2250$:  45T244
$6750$:  45T423
$843750$:  45T1727
$4218750$:  45T2265
$21093750$:  45T2918

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Degree 9: $C_3^2 : C_6$

Degree 15: None

Low degree siblings

45T3793 x 15

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