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Properties

Label 27T101
Degree $27$
Order $243$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_3^3:C_3^2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(27, 101);
 

Group invariants

Abstract group:  $C_3^3:C_3^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $243=3^{5}$
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:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $27$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $101$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,16,22)(2,17,23)(3,18,24)(4,10,25)(5,11,26)(6,12,27)(7,13,19)(8,14,20)(9,15,21)$, $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$, $(4,5,6)(7,9,8)(13,14,15)(16,18,17)(22,23,24)(25,27,26)$, $(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$ x 40
$9$:  $C_3^2$ x 130
$27$:  27T4 x 40

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$

Low degree siblings

27T101 x 39

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

83 x 83 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed