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Properties

Label 42T719
Degree $42$
Order $18522$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:C_3^2:S_3$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(42, 719);
 

Group invariants

Abstract group:  $C_7^3:C_3^2:S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $18522=2 \cdot 3^{3} \cdot 7^{3}$
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 $n$:  $42$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $719$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,31,20,39,23,33,28,42,17,36,22,30,25,38)(16,32,19,40,24,34,27,41,18,35,21,29,26,37)$, $(1,15,35,10,25,34,4,22,32,12,17,29,6,28,41,14,23,40,8,20,37)(2,16,36,9,26,33,3,21,31,11,18,30,5,27,42,13,24,39,7,19,38)$, $(1,32,17,4,37,23,6,29,15,8,35,22,10,41,28,12,34,20,14,40,25)(2,31,18,3,38,24,5,30,16,7,36,21,9,42,27,11,33,19,13,39,26)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$ x 4
$18$:  $C_3^2:C_2$
$54$:  $(C_3^2:C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 7: None

Degree 14: None

Degree 21: 21T65

Low degree siblings

21T65 x 4, 42T719 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

51 x 51 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed