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Properties

Label 40T45488
Degree $40$
Order $115200$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_5^2:D_4$

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Copy content magma:G := TransitiveGroup(40, 45488);
 

Group invariants

Abstract group:  $S_5^2:D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $115200=2^{9} \cdot 3^{2} \cdot 5^{2}$
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 $t$:  $45488$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,19,16,2,20,15)(3,18,10,4,17,9)(5,12,7,6,11,8)(13,14)(21,38,33,30,40,32)(22,37,34,29,39,31)(23,26,28,24,25,27)$, $(1,24,7,37,4,40)(2,23,8,38,3,39)(5,35,19,21,13,30)(6,36,20,22,14,29)(9,26)(10,25)(11,31,17,33,16,28)(12,32,18,34,15,27)$, $(1,34,7,35,5,29,18,23,15,26,2,33,8,36,6,30,17,24,16,25)(3,21,12,38,19,28,9,40,13,32,4,22,11,37,20,27,10,39,14,31)$
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
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$
$28800$:  $S_5^2 \wr C_2$
$57600$:  20T655

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: None

Degree 8: None

Degree 10: None

Degree 20: 20T540

Low degree siblings

20T781 x 8, 24T17918 x 8, 40T45487 x 4, 40T45488 x 7, 40T45489 x 8, 40T45497 x 4, 40T45508 x 4, 40T45542 x 8, 40T45559 x 4, 40T45560 x 4

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