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Properties

Label 36T30524
Degree $36$
Order $559872$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^6.C_2\wr D_6$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 30524);
 

Group invariants

Abstract group:  $C_3^6.C_2\wr D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $559872=2^{8} \cdot 3^{7}$
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 $t$:  $30524$
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:  $(1,10)(2,9)(3,11)(4,12)(5,8)(6,7)(13,17,15,14,18,16)(25,34,28,35,29,31,26,33,27,36,30,32)$, $(1,13)(2,14)(3,16)(4,15)(5,18)(6,17)(7,9)(8,10)(19,25,20,26)(21,30,22,29)(23,27,24,28)(31,33,36,32,34,35)$, $(1,22)(2,21)(3,20,6,24)(4,19,5,23)(7,35,10,31)(8,36,9,32)(11,33,12,34)(13,26,16,29)(14,25,15,30)(17,27,18,28)$
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
$6$:  $S_3$
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 3
$16$:  $D_4\times C_2$
$24$:  $S_4$, $S_3 \times C_2^2$
$48$:  $S_4\times C_2$ x 3, 12T28
$72$:  $C_3^2:D_4$
$96$:  12T48
$144$:  12T77
$192$:  $V_4^2:(S_3\times C_2)$, 12T86
$384$:  12T136
$432$:  12T156
$768$:  12T186
$1728$:  24T4943
$3888$:  18T440
$6912$:  24T9626
$15552$:  36T10082
$34992$:  18T675
$62208$:  36T17293
$139968$:  36T21098

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$

Degree 9: None

Degree 12: 12T136

Degree 18: 18T675

Low degree siblings

36T30523 x 6, 36T30524 x 5, 36T30525 x 6, 36T30526 x 6, 36T30527 x 6, 36T30528 x 6, 36T30529 x 6, 36T30530 x 6

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