Properties

Label 27T434
Degree $27$
Order $2187$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_9\wr C_3$

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Show commands: Magma

magma: G := TransitiveGroup(27, 434);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $434$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_9\wr C_3$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $9$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (10,18,14,11,16,15,12,17,13), (1,14,24)(2,15,23)(3,13,22)(4,17,25)(5,18,27)(6,16,26)(7,10,20)(8,11,19)(9,12,21)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_9$ x 3, $C_3^2$
$27$:  $C_9:C_3$ x 2, $C_3^2:C_3$, 27T2
$81$:  $C_3 \wr C_3 $, 27T17, 27T20, 27T23
$243$:  27T104, 27T108 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 9: $C_3 \wr C_3 $

Low degree siblings

27T434 x 26

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $2187=3^{7}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $5$
Label:  2187.366
magma: IdentifyGroup(G);
 
Character table:    not computed

magma: CharacterTable(G);