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Properties

Label 36T19520
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, 19520);
 

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 $t$:  $19520$
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:  $(1,19,18,33,6,22,15,36,3,23,14,32,2,20,17,34,5,21,16,35,4,24,13,31)(7,25,12,28,10,30,8,26,11,27,9,29)$, $(1,19,4,24,5,21,2,20,3,23,6,22)(7,32,14,25,9,34,17,29,11,35,16,27,8,31,13,26,10,33,18,30,12,36,15,28)$, $(1,34,13,29)(2,33,14,30)(3,31,15,26)(4,32,16,25)(5,36,17,28)(6,35,18,27)(7,21)(8,22)(9,24)(10,23)(11,19)(12,20)$
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: 12T199

Degree 18: 18T421

Low degree siblings

36T19521

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