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Properties

Label 36T104775
Degree $36$
Order $6530347008$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^{12}.C_2^6.D_6.(C_2\times D_4)$

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Copy content magma:G := TransitiveGroup(36, 104775);
 

Group invariants

Abstract group:  $C_3^{12}.C_2^6.D_6.(C_2\times D_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $6530347008=2^{12} \cdot 3^{13}$
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 $t$:  $104775$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,36,3,34)(2,35)(4,15,6,13,5,14)(7,12,9,10)(8,11)(16,19,18,20,17,21)(22,31)(23,33,24,32)(25,28,26,29,27,30)$, $(1,22)(2,24)(3,23)(4,19,6,20,5,21)(7,16,8,17)(9,18)(10,13,12,15,11,14)(25,35,27,34)(26,36)(28,32)(29,33)(30,31)$, $(1,24,15,36,26,10,3,22,14,34,25,11)(2,23,13,35,27,12)(4,31,17,7,28,21,5,33,16,9,30,20,6,32,18,8,29,19)$, $(1,3,2)(4,10,5,12)(6,11)(7,13)(8,14)(9,15)(16,17,18)(19,20,21)(22,28)(23,30,24,29)(25,33,26,32,27,31)(34,36)$
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
$6$:  $S_3$
$8$:  $D_{4}$ x 12, $C_2^3$ x 15
$12$:  $D_{6}$ x 7
$16$:  $D_4\times C_2$ x 18, $C_2^4$
$24$:  $S_4$, $S_3 \times C_2^2$ x 7
$32$:  $C_2^2 \wr C_2$ x 4, $C_2^2 \times D_4$ x 3
$48$:  $S_4\times C_2$ x 7, 12T28 x 6, 24T30
$64$:  16T105
$96$:  12T48 x 7, 24T143 x 3
$192$:  $V_4^2:(S_3\times C_2)$, 12T86 x 6, 24T360, 24T400
$384$:  12T136 x 3, 24T1076 x 3
$768$:  12T186 x 2, 24T2202, 24T2481
$1536$:  24T3109 x 2, 24T4787
$3072$:  16T1519
$6144$:  24T8190
$12288$:  24T10366

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: None

Low degree siblings

36T104850, 36T105908, 36T106329

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