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Properties

Label 42T704
42T704 1 19 1->19 32 1->32 2 20 2->20 31 2->31 3 25 3->25 41 3->41 4 26 4->26 42 4->42 5 17 5->17 38 5->38 6 18 6->18 37 6->37 7 24 7->24 33 7->33 8 23 8->23 34 8->34 9 16 9->16 30 9->30 10 15 10->15 29 10->29 11 22 11->22 39 11->39 12 21 12->21 40 12->40 13 27 13->27 36 13->36 14 28 14->28 35 14->35 15->9 15->20 16->10 16->19 17->14 17->22 18->13 18->21 19->4 19->24 20->3 20->23 21->8 21->25 22->7 22->26 23->12 23->27 24->11 24->28 25->2 25->16 26->1 26->15 27->6 27->17 28->5 28->18 29->2 29->38 30->1 30->37 31->8 31->36 32->7 32->35 33->14 34->13 35->6 36->5 37->12 38->11 39->4 39->42 40->3 40->41 41->9 42->10
Degree $42$
Order $16464$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_7^3:S_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(42, 704);
 

Group invariants

Abstract group:  $C_2\times C_7^3:S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $16464=2^{4} \cdot 3 \cdot 7^{3}$
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 $t$:  $704$
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:  $(1,32,7,33,14,35,6,37,12,40,3,41,9,30)(2,31,8,34,13,36,5,38,11,39,4,42,10,29)(15,20,23,27,17,22,26)(16,19,24,28,18,21,25)$, $(1,19,4,26)(2,20,3,25)(5,17,14,28)(6,18,13,27)(7,24,11,22)(8,23,12,21)(9,16,10,15)(29,38)(30,37)(31,36)(32,35)(39,42)(40,41)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$
$48$:  $S_4\times C_2$
$8232$:  21T46

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4\times C_2$

Degree 7: None

Degree 14: None

Degree 21: 21T46

Low degree siblings

42T703 x 2, 42T704

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

140 x 140 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed