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Properties

Label 36T28688
Degree $36$
Order $373248$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6^4.D_6\wr C_2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 28688);
 

Group invariants

Abstract group:  $C_6^4.D_6\wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $373248=2^{9} \cdot 3^{6}$
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 $t$:  $28688$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,27,16,33,9,20)(2,28,15,34,10,19)(3,25,17,35,8,21)(4,26,18,36,7,22)(5,29,13,32,12,23)(6,30,14,31,11,24)$, $(7,11,10)(8,12,9)(13,18,16,14,17,15)(19,33,20,34)(21,31,22,32)(23,36,24,35)(25,30)(26,29)(27,28)$, $(1,19,13,25,8,32)(2,20,14,26,7,31)(3,24,17,29,12,35,5,22,16,27,9,33)(4,23,18,30,11,36,6,21,15,28,10,34)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$
$72$:  $C_3^2:D_4$ x 2
$144$:  12T77 x 2
$288$:  12T125 x 2
$1152$:  $S_4\wr C_2$
$2304$:  12T235
$2592$:  12T248
$4608$:  12T260
$23328$:  18T648
$41472$:  24T14605

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $C_3^2:D_4$

Degree 9: None

Degree 12: 12T235

Degree 18: 18T648

Low degree siblings

36T28687, 36T28689, 36T28690, 36T28691, 36T28692, 36T28693, 36T28694, 36T28695, 36T28696, 36T28697, 36T28698, 36T28699, 36T28700, 36T28701, 36T28702

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