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Properties

Label 36T120220
Degree $36$
Order $5.642\times 10^{12}$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^{12}.C_2^8.C_3^4.C_2^6.C_2^3$

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Copy content magma:G := TransitiveGroup(36, 120220);
 

Group invariants

Abstract group:  $C_3^{12}.C_2^8.C_3^4.C_2^6.C_2^3$
Copy content magma:IdentifyGroup(G);
 
Order:  $5642219814912=2^{17} \cdot 3^{16}$
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 $t$:  $120220$
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:  $(1,35,15,10,3,36,13,12)(2,34,14,11)(4,33,17,7,29,20,5,31,18,8,30,21,6,32,16,9,28,19)(22,26,23,25)(24,27)$, $(1,8,2,9)(3,7)(4,35,5,36,6,34)(10,30)(11,29)(12,28)(13,33,15,32,14,31)(16,24,17,22,18,23)(19,26)(20,25)(21,27)$, $(1,10,3,11)(2,12)(4,19,29,31,18,7,6,21,30,33,17,8)(5,20,28,32,16,9)(13,36,14,34,15,35)(22,27,23,26,24,25)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 20, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 10, $C_2^2:C_4$ x 12, $Q_8:C_2$ x 4, $C_4\times C_2^2$
$32$:  $C_2^2 \wr C_2$ x 8, $C_2^3 : C_4 $ x 8, $C_4 \times D_4$ x 4, $C_2 \times (C_2^2:C_4)$ x 3, 16T30 x 2, 16T34 x 12, 16T37 x 4, $C_4^2:C_2$ x 2
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 8, 16T76 x 4, 16T79, 32T239 x 4, 32T241 x 2, 32T308 x 4, 32T320 x 4
$128$:  16T208 x 4, 16T230 x 4, 16T240 x 2, 16T350 x 8
$256$:  32T3707 x 2, 32T4130
$512$:  16T903
$5184$:  12T261
$10368$:  24T10061 x 2, 24T10065
$41472$:  24T14539
$1327104$:  16T1936
$2654208$:  24T22629 x 2, 24T22630
$10616832$:  24T23796

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: 12T261

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