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Properties

Label 36T33629
Degree $36$
Order $663552$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^5.A_4^3:A_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 33629);
 

Group invariants

Abstract group:  $C_2^5.A_4^3:A_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $663552=2^{13} \cdot 3^{4}$
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 $t$:  $33629$
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:  $(1,5,2,6)(3,8,4,7)(9,12)(10,11)(17,18)(19,20)(25,33,26,34)(27,36,28,35)(29,31)(30,32)$, $(1,27,20,7,34,22,12,29,13,2,28,19,8,33,21,11,30,14)(3,25,18,6,35,24,9,31,16,4,26,17,5,36,23,10,32,15)$, $(1,11,8,2,12,7)(3,10,5,4,9,6)(13,23,14,24)(15,22,16,21)(17,20)(18,19)(25,29,26,30)(27,31,28,32)(33,36)(34,35)$, $(1,22,34,8,13,28,12,19,29)(2,21,33,7,14,27,11,20,30)(3,24,35,5,16,26,9,17,31)(4,23,36,6,15,25,10,18,32)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$ x 21
$24$:  $A_4\times C_2$ x 21
$48$:  $C_2^4:C_3$ x 21
$96$:  12T56 x 21
$192$:  24T390
$324$:  $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$
$384$:  24T1022
$648$:  18T199
$1296$:  18T322 x 5
$2592$:  18T400 x 5
$5184$:  36T6050
$10368$:  36T8827
$20736$:  12T284
$41472$:  18T699
$82944$:  18T786 x 5
$165888$:  18T839 x 5
$331776$:  36T28369

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$

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

Degree 12: None

Degree 18: 18T322, 18T839 x 2

Low degree siblings

36T33629 x 1439, 36T33721 x 2880

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