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Properties

Label 45T966
45T966 1 2 1->2 27 1->27 26 2->26 3 25 3->25 4 4->26 29 4->29 5 5->25 28 5->28 6 6->27 30 6->30 7 15 7->15 23 7->23 8 14 8->14 22 8->22 9 13 9->13 24 9->24 10 18 10->18 37 10->37 11 17 11->17 38 11->38 12 16 12->16 39 12->39 13->17 13->38 14->16 14->37 15->18 15->39 16->5 32 16->32 17->6 31 17->31 18->4 33 18->33 19 19->8 19->29 20 20->9 20->30 21 21->7 21->28 22->1 23->2 23->9 24->3 24->7 41 25->41 42 25->42 40 26->40 26->40 27->41 27->42 43 28->43 44 29->44 45 30->45 31->12 31->44 32->11 32->43 33->10 33->45 34 34->6 34->31 35 35->5 35->32 36 36->4 36->33 37->35 38->36 39->34 40->21 40->36 41->20 41->35 42->19 42->34 43->14 43->22 44->13 44->24 45->15 45->23
Degree $45$
Order $81000$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{15}^3.S_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 966);
 

Group invariants

Abstract group:  $C_{15}^3.S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $81000=2^{3} \cdot 3^{4} \cdot 5^{3}$
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 $n$:  $45$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $966$
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:  $(1,27,41,20,9,13,38,36,33,10,18,4,29,44,24,3,25,42,19,8,14,37,35,32,11,17,6,30,45,23,2,26,40,21,7,15,39,34,31,12,16,5,28,43,22)$, $(1,2)(4,26,40,36)(5,25,41,35)(6,27,42,34)(7,23,9,24)(8,22)(10,37)(11,38)(12,39)(13,17,31,44)(14,16,32,43)(15,18,33,45)(19,29)(20,30)(21,28)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$
$648$:  $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
$3000$:  15T51

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Degree 9: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

Degree 15: 15T51

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