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Properties

Label 36T19521
Degree $36$
Order $93312$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^5:(\He_3^2:C_4)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 19521);
 

Group invariants

Abstract group:  $C_2^5:(\He_3^2:C_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $93312=2^{7} \cdot 3^{6}$
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 $t$:  $19521$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $6$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,33,4,31,5,36,2,34,3,32,6,35)(7,29,17,24,10,27,16,19,11,26,14,21)(8,30,18,23,9,28,15,20,12,25,13,22)$, $(1,27,2,28)(3,29,4,30)(5,26,6,25)(7,31,15,20)(8,32,16,19)(9,33,14,21)(10,34,13,22)(11,35,18,23)(12,36,17,24)$, $(1,31,8,22,3,35,12,20,5,34,9,23)(2,32,7,21,4,36,11,19,6,33,10,24)(13,27,17,25,15,29,14,28,18,26,16,30)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$36$:  $C_3^2:C_4$ x 10
$72$:  12T40 x 10
$324$:  18T128
$576$:  $A_4^2:C_4$
$648$:  36T1029
$972$:  27T276, 27T283
$1152$:  12T198
$2916$:  18T421
$5184$:  24T7772
$5832$:  36T6367
$10368$:  36T8404
$46656$:  36T15897

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $C_3^2:C_4$

Degree 9: None

Degree 12: 12T198

Degree 18: 18T421

Low degree siblings

36T19520

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

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed