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Properties

Label 42T905
42T905 1 3 1->3 28 1->28 2 4 2->4 27 2->27 19 3->19 20 4->20 5 14 5->14 25 5->25 6 13 6->13 26 6->26 7 11 7->11 17 7->17 8 12 8->12 18 8->18 9 10 9->10 23 9->23 24 10->24 15 11->15 16 12->16 22 13->22 21 14->21 29 15->29 36 15->36 30 16->30 35 16->35 17->36 41 17->41 18->35 42 18->42 34 19->34 19->41 33 20->33 20->42 21->33 39 21->39 22->34 40 22->40 32 23->32 23->40 31 24->31 24->39 25->32 38 25->38 26->31 37 26->37 27->30 27->38 28->29 28->37 29->1 29->22 30->2 30->21 31->10 31->25 32->9 32->26 33->4 33->16 34->3 34->15 35->12 35->20 36->11 36->19 37->6 37->23 38->5 38->24 39->14 39->27 40->13 40->28 41->7 41->17 42->8 42->18
Degree $42$
Order $32928$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_7\times C_{14}^2):S_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(42, 905);
 

Group invariants

Abstract group:  $(C_7\times C_{14}^2):S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $32928=2^{5} \cdot 3 \cdot 7^{3}$
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 $t$:  $905$
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:  $(1,3)(2,4)(5,14)(6,13)(7,11)(8,12)(9,10)(15,29,22,34)(16,30,21,33)(17,36,19,41)(18,35,20,42)(23,40,28,37)(24,39,27,38)(25,32,26,31)$, $(1,28,29)(2,27,30)(3,19,34)(4,20,33)(5,25,38)(6,26,37)(7,17,41)(8,18,42)(9,23,32)(10,24,31)(11,15,36)(12,16,35)(13,22,40)(14,21,39)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$ x 3
$96$:  $V_4^2:S_3$
$8232$:  21T46

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

Degree 7: None

Degree 14: None

Degree 21: 21T46

Low degree siblings

42T905, 42T906 x 2

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

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed