↑

Properties

Label 36T3967
Degree $36$
Order $2592$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:D_6\times S_4$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 3967);
 

Group invariants

Abstract group:  $C_3^2:D_6\times S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $2592=2^{5} \cdot 3^{4}$
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 $t$:  $3967$
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:  $(1,25,15,3,27,13)(2,26,14)(4,11,16,22,30,35)(5,10,17,24,29,34)(6,12,18,23,28,36)(7,32,20,8,31,21)(9,33,19)$, $(1,18,20,26,29,8)(2,16,21,25,28,9)(3,17,19,27,30,7)(4,33,13)(5,31,15)(6,32,14)(10,36,12,35,11,34)(22,23,24)$, $(1,11,21,3,12,20)(2,10,19)(4,5)(7,26,35,9,25,34)(8,27,36)(13,22,31)(14,23,33,15,24,32)(17,18)(28,30)$
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
$6$:  $S_3$ x 3
$8$:  $C_2^3$
$12$:  $D_{6}$ x 9
$24$:  $S_4$, $S_3 \times C_2^2$ x 3
$36$:  $S_3^2$ x 3
$48$:  $S_4\times C_2$ x 3
$72$:  12T37 x 3
$96$:  12T48
$108$:  $C_3^2 : D_{6} $
$144$:  12T83 x 2
$216$:  12T117, 18T94
$288$:  18T111 x 2
$648$:  18T191
$864$:  24T2661

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: $S_4$

Degree 6: None

Degree 9: $C_3^2 : D_{6} $

Degree 12: 12T83

Degree 18: None

Low degree siblings

36T3967

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

55 x 55 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed