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Properties

Label 36T1542
Degree $36$
Order $972$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:S_3^2$

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Copy content magma:G := TransitiveGroup(36, 1542);
 

Group invariants

Abstract group:  $C_3^3:S_3^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $972=2^{2} \cdot 3^{5}$
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 $t$:  $1542$
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:  $(1,20,3,21,2,19)(4,10)(5,11)(6,12)(7,15,8,14,9,13)(16,36)(17,35)(18,34)(22,30)(23,28)(24,29)(25,32,27,31,26,33)$, $(1,29)(2,28)(3,30)(4,27)(5,25)(6,26)(7,22,9,24,8,23)(10,20,11,19,12,21)(13,16)(14,18)(15,17)(31,36,32,34,33,35)$, $(1,30)(2,29)(3,28)(4,27)(5,25)(6,26)(7,23,8,24,9,22)(10,20,12,21,11,19)(13,18)(14,17)(15,16)(31,36,33,35,32,34)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 6
$12$:  $D_{6}$ x 6
$18$:  $C_3^2:C_2$
$36$:  $S_3^2$ x 9, 18T12
$54$:  $(C_3^2:C_3):C_2$
$108$:  12T71 x 4, 18T52, 18T58 x 2
$324$:  18T135 x 2, 36T521

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: $D_6$

Degree 18: None

Low degree siblings

36T1542 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

51 x 51 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed