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Properties

Label 36T13453
Degree $36$
Order $31104$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_2^2\times C_6^2).S_3^3$

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Copy content magma:G := TransitiveGroup(36, 13453);
 

Group invariants

Abstract group:  $(C_2^2\times C_6^2).S_3^3$
Copy content magma:IdentifyGroup(G);
 
Order:  $31104=2^{7} \cdot 3^{5}$
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 $t$:  $13453$
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:  $(1,28,4,29,5,26,2,27,3,30,6,25)(7,23,11,20,10,21)(8,24,12,19,9,22)(13,31,16,34,17,35,14,32,15,33,18,36)$, $(1,14,7,2,13,8)(3,18,11,6,15,9)(4,17,12,5,16,10)(19,25,32,20,26,31)(21,30,33,24,27,36)(22,29,34,23,28,35)$, $(1,10)(2,9)(3,11)(4,12)(5,7)(6,8)(13,18)(14,17)(15,16)(19,21)(20,22)(23,24)(25,33)(26,34)(27,31)(28,32)(29,35)(30,36)$
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
$8$:  $C_2^3$
$12$:  $D_{6}$ x 9
$24$:  $S_3 \times C_2^2$ x 3
$36$:  $S_3^2$ x 3
$72$:  12T37 x 3
$108$:  $C_3^2 : D_{6} $ x 2
$216$:  12T117, 18T94 x 2
$576$:  $(A_4\wr C_2):C_2$
$648$:  18T191 x 2
$1152$:  12T195
$1944$:  18T342
$3456$:  24T7230
$10368$:  36T9018 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $S_3^2$

Degree 9: None

Degree 12: 12T195

Degree 18: 18T343

Low degree siblings

36T13454, 36T13455, 36T13456

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

90 x 90 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed