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Properties

Label 36T480
36T480 1 14 1->14 31 1->31 2 13 2->13 32 2->32 3 15 3->15 36 3->36 4 16 4->16 35 4->35 5 17 5->17 33 5->33 6 18 6->18 34 6->34 7 7->3 26 7->26 8 8->4 25 8->25 9 9->1 28 9->28 10 10->2 27 10->27 11 11->5 29 11->29 12 12->6 30 12->30 13->7 24 13->24 14->8 23 14->23 15->12 22 15->22 16->11 21 16->21 17->10 20 17->20 18->9 19 18->19 19->16 19->34 20->15 20->33 21->13 21->35 22->14 22->36 23->17 23->31 24->18 24->32 25->20 26->19 27->22 28->21 29->24 30->23 31->3 31->26 32->4 32->25 33->1 33->28 34->2 34->27 35->6 35->30 36->5 36->29
Degree $36$
Order $324$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_9:C_6^2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 480);
 

Group invariants

Abstract group:  $C_9:C_6^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $324=2^{2} \cdot 3^{4}$
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 $t$:  $480$
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:  $(1,14,8,4,16,11,5,17,10,2,13,7,3,15,12,6,18,9)(19,34,27,22,36,29,24,32,25,20,33,28,21,35,30,23,31,26)$, $(1,31,3,36,5,33)(2,32,4,35,6,34)(7,26)(8,25)(9,28)(10,27)(11,29)(12,30)(13,24,18,19,16,21)(14,23,17,20,15,22)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$ x 4
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 12
$9$:  $C_3^2$
$12$:  $D_{6}$, $C_6\times C_2$ x 4
$18$:  $S_3\times C_3$ x 4, $C_6 \times C_3$ x 3
$36$:  $C_6\times S_3$ x 4, 36T4
$54$:  $(C_9:C_3):C_2$, $C_3^2\times S_3$
$108$:  18T45, 36T64
$162$:  18T83

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 9: None

Degree 12: $D_6$

Degree 18: 18T83

Low degree siblings

There are no siblings 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

60 x 60 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed