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Properties

Label 36T116338
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, 116338);
 

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 $t$:  $116338$
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:  $(1,20,6,13,35,18,7,11,2,19,4,14,36,16,8,10,3,21,5,15,34,17,9,12)(28,30,29)(31,33,32)$, $(1,17,9,12,35,20,5,15,3,18,8,11,36,19,6,13,2,16,7,10,34,21,4,14)(22,29,23,28,24,30)(25,31,26,33,27,32)$, $(1,19,27)(2,21,26,3,20,25)(4,13,29,6,15,30)(5,14,28)(7,10,32,9,11,33,8,12,31)(16,22,35,17,23,36,18,24,34)$
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: 12T147

Degree 18: None

Low degree siblings

36T116232, 36T116542 x 2

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