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Properties

Label 36T15865
Degree $36$
Order $46656$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6^4.S_3^2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 15865);
 

Group invariants

Abstract group:  $C_6^4.S_3^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $46656=2^{6} \cdot 3^{6}$
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 $t$:  $15865$
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:  $(1,11,18,24,28,34,6,9,15,22,29,31,3,8,13,20,26,36)(2,12,17,23,27,33,5,10,16,21,30,32,4,7,14,19,25,35)$, $(3,6)(4,5)(7,33,12,35,10,32)(8,34,11,36,9,31)(13,29,16,25,18,28,14,30,15,26,17,27)(19,20)(21,24)(22,23)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$ x 2, $C_6$ x 3
$12$:  $D_{6}$ x 2, $C_6\times C_2$
$18$:  $S_3\times C_3$ x 2
$24$:  $S_4$
$36$:  $S_3^2$, $C_6\times S_3$ x 2
$48$:  $S_4\times C_2$
$54$:  $(C_9:C_3):C_2$
$72$:  12T45
$108$:  $C_3^2 : D_{6} $, 12T70, 18T45
$144$:  12T83, 18T61
$192$:  $V_4^2:(S_3\times C_2)$
$216$:  18T98
$324$:  $((C_3^3:C_3):C_2):C_2$, 18T118, 18T122
$432$:  18T147, 18T152, 24T1328
$576$:  24T1496, 24T1497
$972$:  18T232, 18T237
$1296$:  18T299, 36T1989, 36T1999
$1728$:  24T4931, 36T2386, 36T2421
$2916$:  18T407
$3888$:  36T4609, 36T4621
$5184$:  36T5606, 36T5619, 36T5648
$11664$:  36T9112
$15552$:  36T9902, 36T9916

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: 12T108

Degree 18: 18T407

Low degree siblings

36T15864 x 6, 36T15865 x 5

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