↑

Properties

Label 36T6138
Degree $36$
Order $5184$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2\times S_3\wr S_3$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 6138);
 

Group invariants

Abstract group:  $C_2^2\times S_3\wr S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $5184=2^{6} \cdot 3^{4}$
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 $t$:  $6138$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,7)(2,8)(3,6)(4,5)(9,22,16,26,19,30)(10,21,15,25,20,29)(11,24,14,28,18,32)(12,23,13,27,17,31)$, $(1,28,3,26)(2,27,4,25)(5,21,33,31)(6,22,34,32)(7,24,36,30)(8,23,35,29)(9,10)(11,12)(13,18)(14,17)(15,19)(16,20)$, $(1,32,20,4,29,18)(2,31,19,3,30,17)(5,25,11,7,28,10)(6,26,12,8,27,9)(13,33,23,16,36,22)(14,34,24,15,35,21)$, $(1,8)(2,7)(3,5)(4,6)(9,27,16,23,19,31)(10,28,15,24,20,32)(11,25,14,21,18,29)(12,26,13,22,17,30)(33,34)(35,36)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$6$:  $S_3$
$8$:  $C_2^3$ x 15
$12$:  $D_{6}$ x 7
$16$:  $C_2^4$
$24$:  $S_4$, $S_3 \times C_2^2$ x 7
$48$:  $S_4\times C_2$ x 7, 24T30
$96$:  12T48 x 7
$192$:  24T400
$1296$:  $S_3\wr S_3$
$2592$:  18T394 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$, $S_4\times C_2$ x 2

Degree 9: $S_3\wr S_3$

Degree 12: 12T48

Degree 18: 18T394 x 2, 18T397

Low degree siblings

36T6138 x 47, 36T6160 x 16

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

88 x 88 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed