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Properties

Label 36T3592
Degree $36$
Order $2592$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_6^2:D_6$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 3592);
 

Group invariants

Abstract group:  $S_3\times C_6^2:D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $2592=2^{5} \cdot 3^{4}$
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 $t$:  $3592$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,3)(2,4)(7,8)(9,11)(10,12)(13,18)(14,17)(15,16)(19,20)(21,23)(22,24)(25,30)(26,29)(31,33)(32,34)(35,36)$, $(1,33)(2,34)(3,32)(4,31)(5,35)(6,36)(7,30,11,27,10,25)(8,29,12,28,9,26)(13,20,17,22,15,23)(14,19,18,21,16,24)$, $(1,6,3)(2,5,4)(7,17,12,14,10,15,8,18,11,13,9,16)(19,33,23,31,21,36,20,34,24,32,22,35)(25,27,30)(26,28,29)$
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
$6$:  $S_3$ x 3
$8$:  $C_2^3$
$12$:  $D_{6}$ x 9
$24$:  $S_4$, $S_3 \times C_2^2$ x 3
$36$:  $S_3^2$ x 3
$48$:  $S_4\times C_2$ x 3
$72$:  12T37 x 3
$96$:  12T48
$108$:  $C_3^2 : D_{6} $
$144$:  12T83 x 2
$216$:  12T117, 18T94
$288$:  18T111 x 2
$432$:  18T152
$648$:  18T191
$864$:  18T228, 24T2661

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$, $S_4$, $S_4\times C_2$

Degree 9: None

Degree 12: $C_2 \times S_4$

Degree 18: 18T191

Low degree siblings

36T3593, 36T3594, 36T3595

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

60 x 60 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed