↑

Properties

Label 37T11
37T11 1 2 1->2 1->2 3 2->3 4 3->4 5 4->5 6 5->6 7 6->7 8 7->8 9 8->9 10 9->10 11 10->11 12 11->12 13 12->13 14 13->14 15 14->15 16 15->16 17 16->17 18 17->18 19 18->19 20 19->20 21 20->21 22 21->22 23 22->23 24 23->24 25 24->25 26 25->26 27 26->27 28 27->28 29 28->29 30 29->30 31 30->31 32 31->32 33 32->33 34 33->34 35 34->35 36 35->36 37 36->37 37->1
Degree $37$
Order $1.376\times 10^{43}$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_{37}$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(37, 11);
 

Group invariants

Abstract group:  $S_{37}$
Copy content magma:IdentifyGroup(G);
 
Order:  $13763753091226345046315979581580902400000000=2^{34} \cdot 3^{17} \cdot 5^{8} \cdot 7^{5} \cdot 11^{3} \cdot 13^{2} \cdot 17^{2} \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37$
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 $n$:  $37$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $11$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2)$, $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - 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

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed