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Properties

Label 36T10487
Degree $36$
Order $17496$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:S_3^3$

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Copy content magma:G := TransitiveGroup(36, 10487);
 

Group invariants

Abstract group:  $C_3^4:S_3^3$
Copy content magma:IdentifyGroup(G);
 
Order:  $17496=2^{3} \cdot 3^{7}$
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 $t$:  $10487$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,8,2,7,3,9)(4,24,5,22,6,23)(10,16,12,17,11,18)(13,32,14,31,15,33)(19,25,21,26,20,27)(28,36,30,34,29,35)$, $(1,19)(2,21)(3,20)(4,24,6,23,5,22)(7,26)(8,25)(9,27)(10,28,11,29,12,30)(13,32)(14,31)(15,33)(16,36,18,35,17,34)$, $(1,10,2,11,3,12)(4,9,5,7,6,8)(13,35)(14,34)(15,36)(16,33,18,32,17,31)(19,28)(20,30)(21,29)(22,25,23,27,24,26)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$3$:  $C_3$
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 4, $C_6$ x 7
$8$:  $C_2^3$
$12$:  $D_{6}$ x 12, $C_6\times C_2$ x 7
$18$:  $S_3\times C_3$ x 4
$24$:  $S_3 \times C_2^2$ x 4, 24T3
$36$:  $S_3^2$ x 6, $C_6\times S_3$ x 12
$72$:  12T37 x 6, 24T68 x 4
$108$:  $C_3^2 : D_{6} $, 12T70 x 6, 12T71
$216$:  12T117 x 3, 18T94, 24T547 x 6, 24T548
$324$:  $((C_3^3:C_3):C_2):C_2$, 12T130, 18T118
$648$:  18T191 x 2, 18T194, 24T1510, 24T1532, 24T1535 x 3, 36T996
$972$:  18T237
$1944$:  18T341 x 2, 24T4965, 36T2669, 36T2823, 36T2854 x 2
$5832$:  36T6486 x 2, 36T6503

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 3

Degree 9: None

Degree 12: $S_3 \times C_2^2$

Degree 18: None

Low degree siblings

36T10487 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