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Properties

Label 36T12049
Degree $36$
Order $20736$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2\times S_3^3:S_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 12049);
 

Group invariants

Abstract group:  $C_2^2\times S_3^3:S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $20736=2^{8} \cdot 3^{4}$
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 $t$:  $12049$
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:  $(1,11,34,17,7,13,2,12,33,18,8,14)(3,9,36,20,6,15,4,10,35,19,5,16)(21,30)(22,29)(23,32)(24,31)(25,26)(27,28)$, $(1,32,18,5,26,16,33,24,11,4,29,20,7,27,13,36,21,10)(2,31,17,6,25,15,34,23,12,3,30,19,8,28,14,35,22,9)$, $(1,23,4,22)(2,24,3,21)(5,30,33,28)(6,29,34,27)(7,31,36,25)(8,32,35,26)(9,18,15,11,19,13)(10,17,16,12,20,14)$, $(1,28,7,23,33,31)(2,27,8,24,34,32)(3,26,6,21,35,29)(4,25,5,22,36,30)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)$
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
$16$:  $C_2^4$
$24$:  $S_4$ x 3, $S_3 \times C_2^2$ x 7
$48$:  $S_4\times C_2$ x 21, 24T30
$96$:  $V_4^2:S_3$, 12T48 x 21
$192$:  12T100 x 7, 24T400 x 3
$384$:  12T139 x 7
$768$:  24T2499
$1296$:  $S_3\wr S_3$
$2592$:  18T394 x 3
$5184$:  18T483, 36T6138
$10368$:  18T556 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4\times C_2$ x 3

Degree 9: $S_3\wr S_3$

Degree 12: 12T139

Degree 18: 18T394, 18T556 x 2

Low degree siblings

36T12049 x 95, 36T12081 x 96

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

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed