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Properties

Label 40T146640
Degree $40$
Order $691200$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_4\times S_5\wr C_2$

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Copy content magma:G := TransitiveGroup(40, 146640);
 

Group invariants

Abstract group:  $S_4\times S_5\wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $691200=2^{10} \cdot 3^{3} \cdot 5^{2}$
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 $n$:  $40$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $146640$
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:  $(1,40,27,16,10,30,20,7,4,38,28,13,12,29,17,5,3,39,26,14,9,31,18,8)(2,37,25,15,11,32,19,6)(21,35,23,36,22,34)(24,33)$, $(1,19,9,3,20,11,4,18,12,2,17,10)(5,24,13,7,23,15,8,21,16,6,22,14)(25,27,28,26)(29,31,32,30)(33,35,36,34)(37,39,40,38)$, $(1,22,36,5,27,40,20,30,10,16,4,21,34,8,28,38,17,29,12,13,3,23,35,7,26,39,18,31,9,14)(2,24,33,6,25,37,19,32,11,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 3
$16$:  $D_4\times C_2$
$24$:  $S_4$
$48$:  $S_4\times C_2$ x 3
$28800$:  $S_5^2 \wr C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 4: $S_4$

Degree 5: None

Degree 8: $S_4\times C_2$

Degree 10: $S_5^2 \wr C_2$

Degree 20: None

Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

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