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Properties

Label 36T33414
Degree $36$
Order $663552$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4^2.S_4^2:D_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 33414);
 

Group invariants

Abstract group:  $A_4^2.S_4^2:D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $663552=2^{13} \cdot 3^{4}$
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 $t$:  $33414$
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:  $(1,11,13,6,8,17)(2,12,14,5,7,18)(3,9,15,4,10,16)(19,34)(20,33)(21,31)(22,32)(23,36,24,35)(25,28,26,27)(29,30)$, $(1,33,14,27,8,21)(2,34,13,28,7,22)(3,31,15,25,10,19,4,32,16,26,9,20)(5,36,18,29,12,23,6,35,17,30,11,24)$, $(1,22,15,31,5,23,14,33,4,20,17,35)(2,21,16,32,6,24,13,34,3,19,18,36)(7,28,9,26,11,30,8,27,10,25,12,29)$
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 6, $C_2^3$
$12$:  $D_{6}$ x 6
$16$:  $D_4\times C_2$ x 3
$24$:  $S_3 \times C_2^2$ x 2, $D_{12}$ x 2, $(C_6\times C_2):C_2$ x 2
$32$:  $C_2^2 \wr C_2$
$36$:  $S_3^2$
$48$:  12T28 x 4, 24T25, 24T29
$72$:  $C_3^2:D_4$, 12T37, 12T38 x 2
$96$:  24T144, 24T145
$144$:  12T77, 12T81 x 2, 24T230
$288$:  12T125, 24T672
$432$:  12T156 x 2
$576$:  $(A_4\wr C_2):C_2$
$864$:  24T2647, 24T2649
$1152$:  $S_4\wr C_2$, 12T195, 12T196 x 2
$1296$:  12T217
$2304$:  12T235, 12T240 x 2, 24T5079
$2592$:  24T5256
$4608$:  12T260, 24T7509
$6912$:  24T9627 x 2
$13824$:  36T9787, 36T9788
$20736$:  24T12608, 24T12610
$41472$:  36T15412, 36T15418
$331776$:  32T2267336

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $S_3^2$, $C_3^2:D_4$

Degree 9: None

Degree 12: None

Degree 18: 18T289

Low degree siblings

36T33414 x 7, 36T33415 x 8

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