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Properties

Label 36T522
Degree $36$
Order $324$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:S_3^2$

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Copy content magma:G := TransitiveGroup(36, 522);
 

Group invariants

Abstract group:  $C_3^2:S_3^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $324=2^{2} \cdot 3^{4}$
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 $t$:  $522$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $9$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,11,26,34,14,24)(2,10,27,36,15,22)(3,12,25,35,13,23)(4,19,28,8,16,33)(5,20,30,9,18,32)(6,21,29,7,17,31)$, $(1,17,3,18,2,16)(4,15,6,14,5,13)(7,12,9,10,8,11)(19,35,21,36,20,34)(22,31,24,32,23,33)(25,29,27,30,26,28)$, $(1,32,3,33,2,31)(4,23)(5,24)(6,22)(7,25,9,27,8,26)(10,16)(11,17)(12,18)(13,21,15,20,14,19)(28,34)(29,35)(30,36)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$ x 5, $C_6$ x 3
$12$:  $D_{6}$ x 5, $C_6\times C_2$
$18$:  $S_3\times C_3$ x 5, $C_3^2:C_2$
$36$:  $S_3^2$ x 4, $C_6\times S_3$ x 5, 18T12
$54$:  18T23
$108$:  12T70 x 4, 18T58, 36T76

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 9: None

Degree 12: $D_6$, 12T70 x 3

Degree 18: None

Low degree siblings

36T522 x 3, 36T531 x 12

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

54 x 54 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed