↑

Properties

Label 36T12821
Degree $36$
Order $26244$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^7.D_6$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 12821);
 

Group invariants

Abstract group:  $C_3^7.D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $26244=2^{2} \cdot 3^{8}$
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 $t$:  $12821$
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:  $(1,30)(2,29)(3,28)(4,25,6,27,5,26)(7,24,8,22,9,23)(10,19,11,21,12,20)(13,16,14,18,15,17)(31,34)(32,35)(33,36)$, $(1,7,2,9,3,8)(4,34)(5,35)(6,36)(10,29)(11,28)(12,30)(13,31,14,32,15,33)(16,24)(17,22)(18,23)(19,25,21,26,20,27)$, $(1,27,15,2,25,13,3,26,14)(4,29,17,6,30,18,5,28,16)(7,33,20,8,32,21,9,31,19)(10,35,23,11,36,22,12,34,24)$, $(1,10,25,35,14,23,3,12,27,34,13,24,2,11,26,36,15,22)(4,20,28,9,17,31,5,19,30,8,16,32,6,21,29,7,18,33)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 14
$12$:  $D_{6}$ x 14
$18$:  $C_3^2:C_2$ x 13
$36$:  $S_3^2$ x 13, 18T12 x 13
$54$:  $(C_3^2:C_3):C_2$ x 4, 27T7
$108$:  $C_3^2 : D_{6} $, 18T52 x 4, 18T58 x 13
$162$:  $(C_3^3:C_3):C_2$ x 3, 18T89 x 4
$324$:  18T133 x 4, 18T134 x 3, 18T135 x 4, 36T492 x 4
$486$:  27T154, 27T165
$972$:  18T244 x 4, 18T257 x 3, 36T1539 x 4
$1458$:  18T333
$2916$:  18T422 x 3, 36T4056, 36T4125 x 4
$8748$:  36T7470

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 9: None

Degree 12: $D_6$

Degree 18: None

Low degree siblings

36T12821 x 26

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