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Properties

Label 36T119996
Degree $36$
Order $2.821\times 10^{12}$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^{12}.C_2^8.C_3^4.C_2^2:D_4.D_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 119996);
 

Group invariants

Abstract group:  $C_3^{12}.C_2^8.C_3^4.C_2^2:D_4.D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $2821109907456=2^{16} \cdot 3^{16}$
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 $t$:  $119996$
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:  $(1,27,15,2,26,14,3,25,13)(4,30,16)(5,28,17)(6,29,18)(7,33)(8,32,9,31)(10,23)(11,22)(12,24)(34,36)$, $(1,35)(2,36)(3,34)(4,7,29,20,5,9,28,19)(6,8,30,21)(10,13,24,26,12,15,22,27,11,14,23,25)(16,33)(17,32,18,31)$, $(1,17,32,22)(2,16,33,24)(3,18,31,23)(4,8,35,26,30,21,12,14)(5,7,34,25,29,20,11,13,6,9,36,27,28,19,10,15)$, $(1,31,25,19,15,9,3,32,26,20,13,8,2,33,27,21,14,7)(4,34,17,10,28,22)(5,35,18,11,29,23)(6,36,16,12,30,24)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $D_{4}$ x 20, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 30, $C_2^4$
$32$:  $C_2^2 \wr C_2$ x 8, $Q_8:C_2^2$ x 2, $C_2^2 \times D_4$ x 5
$64$:  $(C_4^2 : C_2):C_2$ x 4, 16T87, 16T105 x 2, 16T109 x 4
$128$:  16T265 x 2, 32T1237
$256$:  16T511
$5184$:  12T266
$10368$:  24T10113
$20736$:  24T12559
$1327104$:  16T1934
$2654208$:  24T22681
$5308416$:  24T23348

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: None

Degree 9: None

Degree 12: 12T266

Degree 18: None

Low degree siblings

There are no siblings 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