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Properties

Label 36T7547
Degree $36$
Order $8748$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^5.S_3^2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 7547);
 

Group invariants

Abstract group:  $C_3^5.S_3^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $8748=2^{2} \cdot 3^{7}$
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 $t$:  $7547$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,23,15,36,26,11,2,24,13,35,25,12,3,22,14,34,27,10)(4,33,29,19,16,8,6,32,30,20,17,9,5,31,28,21,18,7)$, $(1,19,3,20,2,21)(4,12)(5,10)(6,11)(7,25,8,26,9,27)(13,33,15,32,14,31)(16,24,17,23,18,22)(28,35,30,34,29,36)$
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 3, $C_6$ x 3
$12$:  $D_{6}$ x 3, $C_6\times C_2$
$18$:  $S_3\times C_3$ x 3
$36$:  $S_3^2$ x 3, $C_6\times S_3$ x 3
$108$:  $C_3^2 : D_{6} $ x 3, 12T70 x 3, 12T71
$324$:  12T130, 18T118 x 3, 27T120
$972$:  27T264, 27T265
$2916$:  18T416

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3^2$

Degree 9: None

Degree 12: $S_3^2$

Degree 18: None

Low degree siblings

36T7544 x 3, 36T7547 x 5

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

96 x 96 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed