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Properties

Label 45T4315
Degree $45$
Order $279936000$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $A_6^3.S_3$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 4315);
 

Group invariants

Abstract group:  $A_6^3.S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $279936000=2^{10} \cdot 3^{7} \cdot 5^{3}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
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 $t$:  $4315$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,26,2,22,11,27)(3,28,6,18,9,29)(4,25,12,30,10,20)(5,16,8,23,7,24)(13,21,14,19,15,17)(31,40,34,44,43,42)(32,45,36,37,41,38)(33,39,35)$, $(1,43,13,39,6,32,14,31,4,38)(2,45,9,33,5,41,3,37,10,40)(7,42,12,34,11,44,8,35,15,36)(16,29,25,27)(17,26)(18,20,23,22)(19,21,28,30)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Degree 9: None

Degree 15: None

Low degree siblings

18T970 x 2, 30T4544, 36T84228 x 2, 45T4315

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

119 x 119 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed