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Properties

Label 36T34160
Degree $36$
Order $746496$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^5.D_4^3:S_3$

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Copy content magma:G := TransitiveGroup(36, 34160);
 

Group invariants

Abstract group:  $C_3^5.D_4^3:S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $746496=2^{10} \cdot 3^{6}$
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 $t$:  $34160$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $6$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,27,5,29,3,26,2,28,6,30,4,25)(7,19,10,22,11,24,8,20,9,21,12,23)(13,17,16,14,18,15)(31,34,36)(32,33,35)$, $(1,20,2,19)(3,22,4,21)(5,24,6,23)(7,29,12,26,9,28,8,30,11,25,10,27)(13,16,18)(14,15,17)(31,32)(33,34)(35,36)$, $(1,24,3,19,6,21)(2,23,4,20,5,22)(7,34,11,31,9,36)(8,33,12,32,10,35)(13,29,14,30)(15,27,16,28)(17,26,18,25)$
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 4
$8$:  $C_2^3$
$12$:  $D_{6}$ x 12
$18$:  $C_3^2:C_2$
$24$:  $S_4$ x 3, $S_3 \times C_2^2$ x 4
$36$:  18T12 x 3
$48$:  $S_4\times C_2$ x 9
$54$:  $(C_3^2:C_3):C_2$
$72$:  12T44 x 3, 36T44
$96$:  $V_4^2:S_3$, 12T48 x 3
$108$:  18T52 x 3
$144$:  18T66 x 9
$192$:  12T100 x 3
$216$:  18T107 x 3, 36T237
$288$:  24T699, 36T362 x 3
$384$:  12T139
$432$:  18T156 x 9
$576$:  36T731 x 3
$768$:  16T1055
$864$:  36T1279, 36T1327 x 3
$1152$:  36T1838
$1296$:  $S_3\wr S_3$
$1536$:  24T3386
$1728$:  36T2367 x 3
$2592$:  18T394
$3072$:  12T250
$3456$:  36T4531
$3888$:  18T444
$5184$:  18T483
$7776$:  36T7274
$9216$:  36T7967
$10368$:  18T556
$11664$:  18T583
$15552$:  36T10093
$23328$:  36T12534
$27648$:  36T13248
$31104$:  36T13554
$46656$:  36T16034
$82944$:  36T19228
$93312$:  36T19684

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4\times C_2$

Degree 9: None

Degree 12: 12T250

Degree 18: 18T583

Low degree siblings

36T34160 x 15

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