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Properties

Label 36T90640
Degree $36$
Order $816293376$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^{12}.C_4^2.S_4.C_2^2$

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Copy content magma:G := TransitiveGroup(36, 90640);
 

Group invariants

Abstract group:  $C_3^{12}.C_4^2.S_4.C_2^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $816293376=2^{9} \cdot 3^{13}$
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 $t$:  $90640$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,30,24,6,2,28,23,4)(3,29,22,5)(7,33,14,35,8,31,15,34,9,32,13,36)(10,18,21,25,12,17,19,26)(11,16,20,27)$, $(1,16,26,22,3,18,25,23,2,17,27,24)(4,33,11,9,5,31,12,8)(6,32,10,7)(13,21,35,30,14,20,34,28)(15,19,36,29)$, $(1,15)(2,13,3,14)(4,20)(5,21,6,19)(7,25)(8,27)(9,26)(10,11)(16,32,18,33,17,31)(22,35,23,36)(24,34)$
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
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$24$:  $S_4$ x 3, $S_3 \times C_2^2$
$48$:  $S_4\times C_2$ x 9
$96$:  $V_4^2:S_3$, 12T48 x 3
$192$:  12T100 x 3
$384$:  12T139
$768$:  16T1063
$1536$:  24T3330

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Degree 9: None

Degree 12: $C_2 \times S_4$

Degree 18: None

Low degree siblings

36T90487, 36T91436, 36T91901

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