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Properties

Label 36T116542
Degree $36$
Order $104485552128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^{12}.C_2^6.C_4^2.A_4.(C_2\times D_4)$

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Copy content magma:G := TransitiveGroup(36, 116542);
 

Group invariants

Abstract group:  $C_3^{12}.C_2^6.C_4^2.A_4.(C_2\times D_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $104485552128=2^{16} \cdot 3^{13}$
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 $t$:  $116542$
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:  $(1,26,3,25)(2,27)(4,22)(5,24)(6,23)(7,20)(8,21)(9,19)(10,18,28,35,12,17,29,36)(11,16,30,34)(13,31,14,32,15,33)$, $(1,17,3,18,2,16)(4,6,5)(7,11,9,12,8,10)(13,33,15,32,14,31)(19,34)(20,35)(21,36)(22,24,23)(25,28)(26,29)(27,30)$, $(1,17,15,28,7,22)(2,16,13,29,9,23,3,18,14,30,8,24)(4,21,35,33,10,26,6,20,34,32,11,27)(5,19,36,31,12,25)$
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
$16$:  $D_4\times C_2$
$24$:  $S_4$ x 3, $S_3 \times C_2^2$, $(C_6\times C_2):C_2$ x 2
$48$:  $S_4\times C_2$ x 9, 24T25
$96$:  $V_4^2:S_3$, 12T48 x 3, 12T49 x 6
$192$:  12T100 x 3, 12T112 x 2, 24T398 x 3
$384$:  12T135 x 2, 12T138 x 2, 12T139, 12T147 x 4, 16T751 x 4
$768$:  16T1063 x 2, 24T1631, 24T1932, 24T2496 x 2, 32T34928 x 2
$1536$:  12T223 x 4, 24T3096, 24T3102 x 2, 24T3330 x 2, 32T97128
$3072$:  24T5377 x 2, 24T6770, 24T7168 x 2
$6144$:  32T397318
$24576$:  24T13910
$196608$:  24T18675

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4\times C_2$

Degree 9: None

Degree 12: 12T223

Degree 18: None

Low degree siblings

36T116232, 36T116338, 36T116542

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