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Properties

Label 36T119930
Degree $36$
Order $2.821\times 10^{12}$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^{12}.C_2^8.C_3^4.C_2^6.C_2^2$

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Copy content magma:G := TransitiveGroup(36, 119930);
 

Group invariants

Abstract group:  $C_3^{12}.C_2^8.C_3^4.C_2^6.C_2^2$
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 $t$:  $119930$
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:  $(1,23,25,34,15,12)(2,24,26,36,13,10,3,22,27,35,14,11)(4,19,6,21)(5,20)(7,18,8,17)(9,16)(28,33,30,31)(29,32)$, $(1,27,2,25)(3,26)(4,29,18)(5,28,17,6,30,16)(7,21,8,19,9,20)(10,12)(13,14,15)(22,23)(31,32)(34,35)$, $(1,21,15,7,26,32,3,20,14,9,27,31,2,19,13,8,25,33)(4,23,18,35,30,11)(5,22,16,34,28,12,6,24,17,36,29,10)$, $(1,29,25,16,13,5)(2,30,26,18,15,6)(3,28,27,17,14,4)(7,11,20,35,32,23,8,12,21,34,33,24)(9,10,19,36,31,22)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_4$ x 8, $C_2^2$ x 35
$8$:  $D_{4}$ x 8, $C_4\times C_2$ x 28, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 12, $C_2^2:C_4$ x 16, $C_4\times C_2^2$ x 14, $C_2^4$
$32$:  $C_2^3 : C_4 $ x 8, $Q_8:C_2^2$ x 4, $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34
$64$:  16T68 x 2, 16T76 x 12, 16T87 x 4, 32T262
$128$:  32T992, 32T1107 x 2
$256$:  16T470
$2592$:  12T242
$5184$:  24T7639 x 2, 24T7644
$20736$:  24T12494
$663552$:  16T1921
$1327104$:  24T21661, 24T21663 x 2
$5308416$:  24T23282

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: None

Degree 9: None

Degree 12: 12T242

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