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Properties

Label 36T70179
Degree $36$
Order $31850496$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^{12}.(C_6^3.S_3^2)$

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Copy content magma:G := TransitiveGroup(36, 70179);
 

Group invariants

Abstract group:  $C_2^{12}.(C_6^3.S_3^2)$
Copy content magma:IdentifyGroup(G);
 
Order:  $31850496=2^{17} \cdot 3^{5}$
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 $t$:  $70179$
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:  $(1,12,5,9,4,8,2,11,6,10,3,7)(13,16,14,15)(19,23)(20,24)(21,22)(25,36)(26,35)(27,31)(28,32)(29,34,30,33)$, $(1,22,8,28,18,36,6,20,9,30,16,33,4,23,12,25,14,31)(2,21,7,27,17,35,5,19,10,29,15,34,3,24,11,26,13,32)$, $(1,2)(3,6,4,5)(7,15,11,13,9,18,8,16,12,14,10,17)(19,36,24,31,22,34,20,35,23,32,21,33)(25,30,26,29)(27,28)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$3$:  $C_3$
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 2, $C_6$ x 7
$8$:  $C_2^3$
$12$:  $D_{6}$ x 6, $C_6\times C_2$ x 7
$18$:  $S_3\times C_3$ x 2
$24$:  $S_4$, $S_3 \times C_2^2$ x 2, 24T3
$36$:  $S_3^2$, $C_6\times S_3$ x 6
$48$:  $S_4\times C_2$ x 3
$54$:  $(C_9:C_3):C_2$
$72$:  12T37, 12T45, 24T68 x 2
$96$:  12T48
$108$:  $C_3^2 : D_{6} $, 12T70, 18T45 x 3
$144$:  12T83, 18T61 x 3
$216$:  18T94, 18T98, 24T547, 36T217
$288$:  18T111, 36T330
$324$:  18T118, 18T122
$432$:  18T147 x 3, 18T152, 24T1328
$648$:  36T996, 36T1013
$864$:  18T228, 36T1297, 36T1298
$972$:  18T232
$1296$:  36T1989, 36T1999
$1944$:  36T2655
$2592$:  36T3513, 36T3527
$3888$:  36T4609
$7776$:  36T7106
$3981312$:  24T23152
$7962624$:  36T57890
$15925248$:  36T64287

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: None

Degree 18: 18T232

Low degree siblings

36T70179

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