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Properties

Label 18T725
18T725 1 9 1->9 15 1->15 2 8 2->8 14 2->14 3 7 3->7 13 3->13 4 4->2 17 4->17 5 5->1 18 5->18 6 6->3 16 6->16 7->4 12 7->12 8->6 11 8->11 9->5 10 9->10 10->3 11->1 12->2 13->6 13->14 14->4 14->15 15->5 15->13 16->7 17->9 18->8
Degree $18$
Order $52488$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\He_3^2:\SOPlus(4,2)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(18, 725);
 

Group invariants

Abstract group:  $\He_3^2:\SOPlus(4,2)$
Copy content magma:IdentifyGroup(G);
 
Order:  $52488=2^{3} \cdot 3^{8}$
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 $n$:  $18$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $725$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,9,5)(2,8,6,3,7,4)(13,14,15)$, $(1,15,5,18,8,11)(2,14,4,17,9,10,3,13,6,16,7,12)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$8$:  $D_{4}$
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$24$:  $(C_6\times C_2):C_2$, $D_4 \times C_3$
$36$:  $C_6\times S_3$
$72$:  $C_3^2:D_4$, 12T42
$216$:  12T116, 12T121
$648$:  12T167
$5832$:  18T492

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $S_3\times C_3$

Degree 9: None

Low degree siblings

18T725 x 2, 36T16230 x 3, 36T16231 x 3, 36T16232 x 3, 36T16478 x 3, 36T16635 x 3

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

104 x 104 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed