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Properties

Label 36T12001
Degree $36$
Order $20736$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6:D_6^2:S_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 12001);
 

Group invariants

Abstract group:  $C_6:D_6^2:S_4$
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 $t$:  $12001$
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:  $(1,22,5,31,34,26)(2,21,6,32,33,25)(3,23,8,29,35,27,4,24,7,30,36,28)(9,20)(10,19)(11,17)(12,18)(15,16)$, $(1,17,32,7,9,27,34,14,21,3,19,30,5,11,25,35,16,23)(2,18,31,8,10,28,33,13,22,4,20,29,6,12,26,36,15,24)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$ x 3
$48$:  $S_4\times C_2$ x 3
$96$:  $V_4^2:S_3$
$192$:  $V_4^2:(S_3\times C_2)$ x 2, 12T100
$648$:  $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
$768$:  16T1068
$1296$:  18T301
$2592$:  18T402
$5184$:  18T485

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$

Degree 9: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

Degree 12: 12T108

Degree 18: 18T313

Low degree siblings

36T12000 x 4, 36T12001 x 3, 36T12166 x 4, 36T12169 x 4

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