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Properties

Label 36T11793
Degree $36$
Order $20736$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6^2:\POPlus(4,3)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 11793);
 

Group invariants

Abstract group:  $C_6^2:\POPlus(4,3)$
Copy content magma:IdentifyGroup(G);
 
Order:  $20736=2^{8} \cdot 3^{4}$
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 $t$:  $11793$
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:  $(1,31,4,36,6,33)(2,32,3,35,5,34)(7,26,10,30,12,27)(8,25,9,29,11,28)(13,19,15,24,17,21)(14,20,16,23,18,22)$, $(1,31,15,27,11,23)(2,32,16,28,12,24)(3,34,18,29,7,19)(4,33,17,30,8,20)(5,35,14,25,10,21)(6,36,13,26,9,22)$, $(1,23,5,21,4,20,2,24,6,22,3,19)(7,29,12,28,10,25)(8,30,11,27,9,26)(13,36,17,33,15,31)(14,35,18,34,16,32)$
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 8
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 24
$16$:  $D_4\times C_2$
$18$:  $C_3^2:C_2$ x 2
$24$:  $S_3 \times C_2^2$ x 8
$36$:  $S_3^2$ x 16, 18T12 x 6
$48$:  12T28 x 8
$72$:  12T37 x 16, 36T44 x 2
$108$:  18T58 x 8
$144$:  12T81 x 16, 36T132 x 2
$216$:  36T255 x 8
$324$:  18T138
$432$:  36T631 x 8
$576$:  $(A_4\wr C_2):C_2$
$648$:  36T1063
$1152$:  12T195
$1296$:  36T2030
$1728$:  24T4933 x 2
$2304$:  12T240
$3456$:  36T4357 x 2
$5184$:  24T7770
$6912$:  36T6899 x 2
$10368$:  36T8438

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $S_3^2$ x 4

Degree 9: None

Degree 12: 12T240

Degree 18: 18T138

Low degree siblings

36T11793 x 23

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