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Properties

Label 36T17208
Degree $36$
Order $62208$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6^3.(D_6\times S_4)$

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Copy content magma:G := TransitiveGroup(36, 17208);
 

Group invariants

Abstract group:  $C_6^3.(D_6\times S_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $62208=2^{8} \cdot 3^{5}$
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 $t$:  $17208$
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:  $(1,14)(2,13)(3,16)(4,15)(5,18)(6,17)(7,24,11,19,9,21,8,23,12,20,10,22)(25,31,28,35,30,34,26,32,27,36,29,33)$, $(1,32,20,8,29,14,5,35,21,10,28,16,3,33,24,11,26,18)(2,31,19,7,30,13,6,36,22,9,27,15,4,34,23,12,25,17)$, $(1,5,3)(2,6,4)(7,11,9,8,12,10)(13,34,17,36,15,31)(14,33,18,35,16,32)(19,29)(20,30)(21,27)(22,28)(23,26)(24,25)$
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 3
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 9
$16$:  $D_4\times C_2$
$24$:  $S_4$, $S_3 \times C_2^2$ x 3
$36$:  $S_3^2$ x 3
$48$:  $S_4\times C_2$ x 3, 12T28 x 3
$72$:  12T37 x 3
$96$:  12T48
$108$:  $C_3^2 : D_{6} $
$144$:  12T81 x 2, 12T83 x 2, 24T231
$192$:  $V_4^2:(S_3\times C_2)$, 12T86
$216$:  12T117, 18T94
$288$:  18T111 x 2
$324$:  $((C_3^3:C_3):C_2):C_2$
$384$:  12T136
$432$:  18T152, 24T1294, 36T625
$576$:  36T761, 36T804
$648$:  18T191, 18T194
$768$:  12T186
$864$:  18T228, 24T2661
$1152$:  24T2820 x 2
$1296$:  18T299, 36T2024
$1728$:  36T2428, 36T2467
$1944$:  18T341
$2304$:  36T3148, 36T3153
$2592$:  18T396, 36T3592
$3456$:  24T7226, 36T4376
$3888$:  36T4691
$5184$:  36T5647
$6912$:  36T6918, 36T6939
$7776$:  36T7181
$10368$:  36T8467, 36T8476
$15552$:  36T9992
$20736$:  36T11833
$31104$:  36T13443

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: 12T193

Degree 18: 18T341

Low degree siblings

36T17208 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