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Properties

Label 36T8404
Degree $36$
Order $10368$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^5:(C_3^4:C_4)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 8404);
 

Group invariants

Abstract group:  $C_2^5:(C_3^4:C_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $10368=2^{7} \cdot 3^{4}$
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 $t$:  $8404$
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:  $(1,2)(3,4)(7,8)(9,10)(13,14)(15,16)(19,28,36)(20,27,35)(21,30,31)(22,29,32)(23,25,33)(24,26,34)$, $(1,21,4,23)(2,22,3,24)(5,19,6,20)(7,28,16,35)(8,27,15,36)(9,29,14,34)(10,30,13,33)(11,25,18,31)(12,26,17,32)$, $(1,19,14,31)(2,20,13,32)(3,22,17,35,4,21,18,36)(5,24,15,33,6,23,16,34)(7,26)(8,25)(9,28,12,30,10,27,11,29)$
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
$1152$:  12T198
$5184$:  24T7772

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $C_3^2:C_4$ x 4

Degree 9: None

Degree 12: 12T198

Degree 18: 18T128

Low degree siblings

36T8404 x 11, 36T8405 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

90 x 90 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed