↑

Properties

Label 42T20
42T20 1 5 1->5 2 4 2->4 3 6 3->6 4->5 9 4->9 5->6 7 5->7 6->4 8 6->8 12 7->12 11 8->11 10 9->10 10->12 15 10->15 11->10 14 11->14 12->11 13 12->13 18 13->18 17 14->17 16 15->16 16->18 20 16->20 17->16 19 17->19 18->17 21 18->21 24 19->24 23 20->23 22 21->22 22->24 26 22->26 23->22 25 23->25 24->23 27 24->27 30 25->30 29 26->29 28 27->28 28->30 31 28->31 29->28 33 29->33 30->29 32 30->32 35 31->35 36 32->36 34 33->34 34->35 37 34->37 35->36 38 35->38 36->34 39 36->39 40 37->40 41 38->41 42 39->42 40->3 40->41 41->2 41->42 42->1 42->40
Degree $42$
Order $126$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_{21}$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(42, 20);
 

Group invariants

Abstract group:  $S_3\times C_{21}$
Copy content magma:IdentifyGroup(G);
 
Order:  $126=2 \cdot 3^{2} \cdot 7$
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 $t$:  $20$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $21$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(4,5,6)(10,12,11)(16,18,17)(22,24,23)(28,30,29)(34,35,36)(40,41,42)$, $(1,5,7,12,13,18,21,22,26,29,33,34,37,40,3,6,8,11,14,17,19,24,27,28,31,35,38,41,2,4,9,10,15,16,20,23,25,30,32,36,39,42)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$7$:  $C_7$
$14$:  $C_{14}$
$18$:  $S_3\times C_3$
$21$:  $C_{21}$
$42$:  21T6, $C_{42}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $S_3\times C_3$

Degree 7: $C_7$

Degree 14: $C_{14}$

Degree 21: None

Low degree siblings

There are no siblings 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

63 x 63 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed