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Properties

Label 36T3524
Degree $36$
Order $2592$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_6^3:C_6$

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Copy content magma:G := TransitiveGroup(36, 3524);
 

Group invariants

Abstract group:  $C_2\times C_6^3:C_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $2592=2^{5} \cdot 3^{4}$
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 $t$:  $3524$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,15,30,2,16,29)(3,13,32,4,14,31)(5,20,23,6,19,24)(7,18,21,8,17,22)(9,27,34,10,28,33)(11,26,35,12,25,36)$, $(1,14,29,4,15,32)(2,13,30,3,16,31)(5,12,24,35,20,26)(6,11,23,36,19,25)(7,9,22,33,18,28)(8,10,21,34,17,27)$, $(1,15,28)(2,16,27)(3,13,25)(4,14,26)(5,9,29,33,20,24)(6,10,30,34,19,23)(7,12,32,35,18,22)(8,11,31,36,17,21)$
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
$6$:  $S_3$, $C_6$ x 7
$8$:  $C_2^3$
$12$:  $A_4$, $D_{6}$ x 3, $C_6\times C_2$ x 7
$18$:  $S_3\times C_3$
$24$:  $A_4\times C_2$ x 7, $S_3 \times C_2^2$, 24T3
$36$:  $C_6\times S_3$ x 3
$48$:  $C_2^2 \times A_4$ x 7
$54$:  $C_3^2 : C_6$
$72$:  12T43, 24T68
$96$:  24T135
$108$:  18T41 x 3
$144$:  18T60 x 3
$162$:  $(C_3^3:C_3):C_2$
$216$:  18T100, 36T204
$288$:  36T334
$324$:  18T125 x 3
$432$:  18T148 x 3
$648$:  18T200, 36T1023
$864$:  36T1288
$1296$:  18T282 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$, $A_4\times C_2$ x 2

Degree 9: $(C_3^3:C_3):C_2$

Degree 12: $C_2^2 \times A_4$

Degree 18: 18T125, 18T282 x 2

Low degree siblings

36T3524 x 107, 36T3725 x 72

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

112 x 112 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed