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Properties

Label 36T19529
Degree $36$
Order $93312$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_6^4.S_3^2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 19529);
 

Group invariants

Abstract group:  $C_2\times C_6^4.S_3^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $93312=2^{7} \cdot 3^{6}$
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 $t$:  $19529$
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:  $(1,20,6,24,3,21)(2,19,5,23,4,22)(7,15,10,17,12,13,8,16,9,18,11,14)(25,35,26,36)(27,33,28,34)(29,31,30,32)$, $(1,36,26,21,15,8)(2,35,25,22,16,7)(3,32,30,24,18,10)(4,31,29,23,17,9)(5,34,28,19,14,12)(6,33,27,20,13,11)$, $(1,35,27,23,18,9)(2,36,28,24,17,10)(3,31,26,19,13,12)(4,32,25,20,14,11)(5,33,29,21,16,8)(6,34,30,22,15,7)$
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 5
$8$:  $C_2^3$
$12$:  $D_{6}$ x 15
$18$:  $C_3^2:C_2$
$24$:  $S_4$, $S_3 \times C_2^2$ x 5
$36$:  $S_3^2$ x 4, 18T12 x 3
$48$:  $S_4\times C_2$ x 3
$54$:  $(C_3^2:C_3):C_2$
$72$:  12T37 x 4, 12T44, 36T44
$96$:  12T48
$108$:  $C_3^2 : D_{6} $, 18T52 x 3, 18T58
$144$:  12T83, 18T66 x 3
$192$:  $V_4^2:(S_3\times C_2)$
$216$:  18T94, 18T107, 36T237, 36T255
$288$:  18T111, 36T362
$324$:  $((C_3^3:C_3):C_2):C_2$, 18T133, 18T135
$384$:  12T136
$432$:  18T152, 18T156 x 3, 24T1329
$576$:  24T1496, 24T1500
$648$:  18T194, 36T1043, 36T1052
$864$:  18T228, 36T1327, 36T1342
$972$:  18T244, 18T255
$1152$:  36T1654, 36T1684
$1296$:  18T299, 36T2022, 36T2032
$1728$:  24T4932, 36T2409, 36T2421
$1944$:  36T2694, 36T2728
$2592$:  18T396, 36T3556, 36T3569
$2916$:  18T424
$3456$:  36T4331, 36T4342, 36T4355
$3888$:  36T4643, 36T4655
$5184$:  36T5644, 36T5648, 36T5656
$5832$:  36T6375
$7776$:  36T7142, 36T7155
$10368$:  36T8429, 36T8431, 36T8441
$11664$:  36T9140
$15552$:  36T9947, 36T9961
$23328$:  36T12429
$31104$:  36T13394, 36T13411
$46656$:  36T15904

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: 12T136

Degree 18: 18T424

Low degree siblings

36T19529 x 11

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