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Properties

Label 45T7968
45T7968 1 8 1->8 28 1->28 2 7 2->7 27 2->27 3 6 3->6 26 3->26 4 10 4->10 30 4->30 5 9 5->9 29 5->29 21 6->21 34 6->34 25 7->25 33 7->33 24 8->24 32 8->32 23 9->23 31 9->31 22 10->22 35 10->35 11 12 11->12 41 11->41 45 12->45 13 15 13->15 44 13->44 14 43 14->43 42 15->42 16 16->12 16->43 17 17->11 17->42 18 18->15 18->41 19 19->14 19->45 20 20->13 20->44 21->18 37 21->37 22->17 36 22->36 23->16 40 23->40 24->20 39 24->39 25->19 38 25->38 26->20 26->31 27->19 27->35 28->18 28->34 29->17 29->33 30->16 30->32 31->4 31->36 32->3 32->40 33->2 33->39 34->1 34->38 35->5 35->37 36->4 36->6 37->3 37->10 38->2 38->9 39->1 39->8 40->5 40->7 41->25 41->28 42->24 42->27 43->23 43->26 44->22 44->30 45->21 45->29
Degree $45$
Order $283500000000$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_5^8.A_9.C_2^2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 7968);
 

Group invariants

Abstract group:  $C_5^8.A_9.C_2^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $283500000000=2^{8} \cdot 3^{4} \cdot 5^{9} \cdot 7$
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 $t$:  $7968$
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:  $(1,8,24,20,13,44,30,32,40,5,9,23,16,12,45,29,33,39)(2,7,25,19,14,43,26,31,36,4,10,22,17,11,41,28,34,38)(3,6,21,18,15,42,27,35,37)$, $(1,28,18,41,25,38,9,31,4,30,16,43,23,40,7,33,2,27,19,45,21,37,10,35,5,29,17,42,24,39,8,32,3,26,20,44,22,36,6,34)(11,12)(13,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$362880$:  $S_9$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: None

Degree 5: None

Degree 9: $S_9$

Degree 15: None

Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

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