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Properties

Label 36T6160
Degree $36$
Order $5184$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2\times S_3\wr S_3$

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Copy content magma:G := TransitiveGroup(36, 6160);
 

Group invariants

Abstract group:  $C_2^2\times S_3\wr S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $5184=2^{6} \cdot 3^{4}$
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 $t$:  $6160$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,19)(14,20)(15,18)(16,17)(21,32)(22,31)(23,29)(24,30)(25,28)(26,27)(33,36)(34,35)$, $(1,21,33,25)(2,22,34,26)(3,23,36,28)(4,24,35,27)(5,31)(6,32)(7,29)(8,30)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)$, $(1,24,15,34,28,9)(2,23,16,33,27,10)(3,22,13,35,25,12)(4,21,14,36,26,11)(5,29,17,6,30,18)(7,31,19,8,32,20)$, $(1,34,7,2,33,8)(3,35,6,4,36,5)(9,23)(10,24)(11,22)(12,21)(13,30,18,26)(14,29,17,25)(15,31,19,27)(16,32,20,28)$
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
$6$:  $S_3$
$8$:  $C_2^3$ x 15
$12$:  $D_{6}$ x 7
$16$:  $C_2^4$
$24$:  $S_4$, $S_3 \times C_2^2$ x 7
$48$:  $S_4\times C_2$ x 7, 24T30
$96$:  12T48 x 7
$192$:  24T400
$1296$:  $S_3\wr S_3$
$2592$:  18T394 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 3

Degree 9: $S_3\wr S_3$

Degree 12: $S_3 \times C_2^2$

Degree 18: 18T397 x 3

Low degree siblings

36T6138 x 48, 36T6160 x 15

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

88 x 88 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed