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Properties

Label 36T23881
Degree $36$
Order $186624$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6^4.C_6:D_{12}$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 23881);
 

Group invariants

Abstract group:  $C_6^4.C_6:D_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $186624=2^{8} \cdot 3^{6}$
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 $t$:  $23881$
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:  $(1,20,11,25,18,32)(2,19,12,26,17,31)(3,24,9,28,13,36)(4,23,10,27,14,35)(5,22,7,29,16,33)(6,21,8,30,15,34)$, $(1,15,10,2,16,9)(3,18,7,4,17,8)(5,14,11,6,13,12)(19,32,30,20,31,29)(21,33,27,22,34,28)(23,36,25,24,35,26)$, $(7,17)(8,18)(9,15)(10,16)(11,13)(12,14)(19,27)(20,28)(21,30)(22,29)(23,26)(24,25)(31,36)(32,35)(33,34)$
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
$16$:  $D_4\times C_2$
$24$:  $S_3 \times C_2^2$ x 2, $D_{12}$ x 2, $(C_6\times C_2):C_2$ x 2
$36$:  $S_3^2$
$48$:  24T25, 24T29
$72$:  $C_3^2:D_4$, 12T37, 12T38 x 2
$144$:  12T77, 24T230
$216$:  12T116, 12T118
$432$:  24T1292, 24T1304
$576$:  $(A_4\wr C_2):C_2$
$648$:  12T169
$1152$:  12T195, 12T196 x 2
$1296$:  24T2850
$2304$:  24T5079
$5832$:  18T507
$10368$:  24T10167
$11664$:  36T9205
$20736$:  36T11869
$93312$:  36T19600

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $S_3^2$

Degree 9: None

Degree 12: 12T196

Degree 18: 18T507

Low degree siblings

36T23880 x 6, 36T23881 x 5

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