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Properties

Label 42T3420
42T3420 1 6 1->6 21 1->21 2 5 2->5 22 2->22 3 3->1 19 3->19 4 4->2 20 4->20 11 5->11 24 5->24 12 6->12 23 6->23 7 13 7->13 25 7->25 8 14 8->14 26 8->26 9 10 9->10 18 9->18 17 10->17 11->4 15 11->15 12->3 16 12->16 28 13->28 27 14->27 40 15->40 41 15->41 39 16->39 42 16->42 34 17->34 35 17->35 33 18->33 36 18->36 32 19->32 19->41 31 20->31 20->42 21->35 37 21->37 22->36 38 22->38 30 23->30 23->30 29 24->29 24->29 25->34 25->37 26->33 26->38 27->31 27->39 28->32 28->40 29->6 29->17 30->5 30->18 31->3 31->16 32->4 32->15 33->12 33->28 34->11 34->27 35->9 36->10 37->13 37->20 38->14 38->19 39->2 39->24 40->1 40->23 41->7 41->26 42->8 42->25
Degree $42$
Order $56899584$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2\times \PSL(2,7)\wr S_3$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(42, 3420);
 

Group invariants

Abstract group:  $C_2\times \PSL(2,7)\wr S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $56899584=2^{11} \cdot 3^{4} \cdot 7^{3}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
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 $t$:  $3420$
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:  $(1,6,12,3)(2,5,11,4)(7,13)(8,14)(9,10)(15,40,23,30,18,33,28,32)(16,39,24,29,17,34,27,31)(19,41,26,38)(20,42,25,37)(21,35)(22,36)$, $(1,21,37,13,28,40)(2,22,38,14,27,39)(3,19,32,4,20,31)(5,24,29,6,23,30)(7,25,34,11,15,41)(8,26,33,12,16,42)(9,18,36,10,17,35)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$28449792$:  21T146

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 7: None

Degree 14: None

Degree 21: 21T146

Low degree siblings

42T3420 x 3, 42T3421 x 4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Conjugacy classes not computed

Copy content magma:ConjugacyClasses(G);
 

Character table

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed