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Properties

Label 36T11999
Degree $36$
Order $20736$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(S_3\times D_6^2).S_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 11999);
 

Group invariants

Abstract group:  $(S_3\times D_6^2).S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $20736=2^{8} \cdot 3^{4}$
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 $n$:  $36$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $11999$
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:  $(1,12,8,19)(2,11,7,20)(3,10,5,17)(4,9,6,18)(13,34,14,33)(15,36,16,35)(21,32,28,24,29,25)(22,31,27,23,30,26)$, $(1,34,7,2,33,8)(3,36,6)(4,35,5)(9,29,16,27,17,21,10,30,15,28,18,22)(11,31,14,26,19,23)(12,32,13,25,20,24)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$ x 3
$48$:  $S_4\times C_2$ x 3
$96$:  $V_4^2:S_3$
$192$:  $V_4^2:(S_3\times C_2)$ x 2, 12T100
$768$:  16T1068
$1296$:  $S_3\wr S_3$
$5184$:  18T483

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$

Degree 9: $S_3\wr S_3$

Degree 12: 12T109

Degree 18: 18T312

Low degree siblings

36T11984 x 2, 36T11985 x 2, 36T11998 x 2, 36T11999, 36T12205 x 2, 36T12206 x 2, 36T12207 x 2, 36T12215 x 2

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

82 x 82 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed