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Properties

Label 45T837
45T837 1 16 1->16 22 1->22 2 20 2->20 25 2->25 3 18 3->18 21 3->21 4 17 4->17 23 4->23 5 19 5->19 24 5->24 6 8 6->8 34 6->34 7 7->6 35 7->35 10 8->10 32 8->32 9 31 9->31 10->7 33 10->33 11 11->17 29 11->29 12 12->16 27 12->27 13 13->20 28 13->28 14 14->19 30 14->30 15 15->18 26 15->26 16->4 45 16->45 17->3 42 17->42 18->1 43 18->43 19->5 41 19->41 20->2 44 20->44 21->15 21->32 22->11 22->35 23->13 23->34 24->14 24->33 25->12 25->31 26->24 26->42 27->25 27->43 28->23 28->45 29->22 29->41 30->21 30->44 31->30 37 31->37 32->27 40 32->40 33->28 38 33->38 34->29 39 34->39 35->26 36 35->36 36->10 36->11 37->9 37->13 38->7 38->14 39->6 39->15 40->8 40->12 41->5 41->40 42->4 42->39 43->2 43->36 44->3 44->37 45->1 45->38
Degree $45$
Order $51840$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_5\times C_3^2:\GL(2,3)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 837);
 

Group invariants

Abstract group:  $S_5\times C_3^2:\GL(2,3)$
Copy content magma:IdentifyGroup(G);
 
Order:  $51840=2^{7} \cdot 3^{4} \cdot 5$
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 $n$:  $45$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $837$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,16,45)(2,20,44,3,18,43)(4,17,42)(5,19,41)(6,34,39)(7,35,36,10,33,38)(8,32,40)(9,31,37)(11,29,22)(12,27,25)(13,28,23)(14,30,21,15,26,24)$, $(1,22,35,26,42,39,15,18)(2,25,31,30,44,37,13,20)(3,21,32,27,43,36,11,17)(4,23,34,29,41,40,12,16)(5,24,33,28,45,38,14,19)(6,8,10,7)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$
$48$:  $S_4\times C_2$, $\textrm{GL(2,3)}$ x 2
$96$:  16T188
$120$:  $S_5$
$240$:  $S_5\times C_2$
$432$:  $((C_3^2:Q_8):C_3):C_2$
$720$:  $S_5 \times S_3$
$864$:  18T229
$2880$:  20T264
$5760$:  40T5154

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $S_5$

Degree 9: $((C_3^2:Q_8):C_3):C_2$

Degree 15: 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

77 x 77 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed