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Properties

Label 36T9215
Degree $36$
Order $11664$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times \He_3^2:D_4$

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Copy content magma:G := TransitiveGroup(36, 9215);
 

Group invariants

Abstract group:  $C_2\times \He_3^2:D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $11664=2^{4} \cdot 3^{6}$
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 $t$:  $9215$
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:  $(1,8,17,6,12,16)(2,7,18,5,11,15)(3,10,13,4,9,14)(19,33)(20,34)(21,32)(22,31)(23,35)(24,36)(25,27)(26,28)$, $(1,34,3,36,5,32,2,33,4,35,6,31)(7,25,16,24,10,27,13,22,12,30,18,19)(8,26,15,23,9,28,14,21,11,29,17,20)$, $(1,6)(2,5)(3,4)(7,18)(8,17)(9,14)(10,13)(11,15)(12,16)(19,28,36,24,26,33)(20,27,35,23,25,34)(21,30,32)(22,29,31)$
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 2, $C_2^3$
$16$:  $D_4\times C_2$
$72$:  $C_3^2:D_4$ x 4
$144$:  12T77 x 4
$648$:  12T172
$1296$:  24T2862
$1944$:  27T403, 27T404
$5832$:  18T509

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: $C_3^2:D_4$

Degree 9: None

Degree 12: 12T78

Degree 18: 18T509

Low degree siblings

36T9214 x 4, 36T9215 x 3

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