↑

Properties

Label 36T2403
Degree $36$
Order $1728$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_2^2\times D_6):D_{18}$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 2403);
 

Group invariants

Abstract group:  $(C_2^2\times D_6):D_{18}$
Copy content magma:IdentifyGroup(G);
 
Order:  $1728=2^{6} \cdot 3^{3}$
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 $t$:  $2403$
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:  $(1,4)(2,3)(7,17,8,18)(9,16,10,15)(11,14,12,13)(19,33,20,34)(21,32,22,31)(23,35,24,36)(27,30)(28,29)$, $(1,28,2,27)(3,26,4,25)(5,30,6,29)(7,23)(8,24)(9,21)(10,22)(11,20)(12,19)(13,36)(14,35)(15,33)(16,34)(17,32)(18,31)$, $(1,35,4,34,6,32)(2,36,3,33,5,31)(7,30,9,27,11,26)(8,29,10,28,12,25)(13,24,15,22,18,19)(14,23,16,21,17,20)$
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 2
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 6
$16$:  $D_4\times C_2$
$18$:  $D_{9}$
$24$:  $S_4$, $S_3 \times C_2^2$ x 2
$36$:  $S_3^2$, $D_{18}$ x 3
$48$:  $S_4\times C_2$ x 3, 12T28 x 2
$72$:  12T37, 18T38, 36T48
$96$:  12T48
$108$:  18T50
$144$:  12T81, 12T83, 18T67 x 3, 36T135
$192$:  12T86
$216$:  36T228
$288$:  18T111, 36T366
$432$:  36T605, 36T606
$576$:  36T761, 36T783
$864$:  36T1313

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: 12T86

Degree 18: 18T50

Low degree siblings

36T2403 x 3

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

75 x 75 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed