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Properties

Label 45T611
45T611 1 16 1->16 17 1->17 2 2->17 18 2->18 3 3->16 3->18 4 4->3 32 4->32 5 5->2 31 5->31 6 6->1 33 6->33 7 7->1 7->4 8 8->3 8->6 9 9->2 9->5 10 26 10->26 27 10->27 11 25 11->25 11->27 12 12->25 12->26 13 13->12 22 13->22 14 14->10 23 14->23 15 15->11 24 15->24 37 16->37 40 16->40 39 17->39 42 17->42 38 18->38 41 18->41 19 34 19->34 35 19->35 20 20->35 36 20->36 21 21->34 21->36 22->20 23->21 24->19 29 25->29 25->31 28 26->28 26->33 30 27->30 27->32 43 28->43 45 28->45 29->43 44 29->44 30->44 30->45 31->28 32->30 33->29 34->23 35->22 36->24 37->7 37->9 38->7 38->8 39->8 39->9 40->38 40->42 41->37 42->39 43->11 43->13 44->10 44->14 45->12 45->15
Degree $45$
Order $18000$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^3:(S_3\times S_4)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 611);
 

Group invariants

Abstract group:  $C_5^3:(S_3\times S_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $18000=2^{4} \cdot 3^{2} \cdot 5^{3}$
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 $t$:  $611$
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:  $(1,17,42,39,8,6)(2,18,41,37,9,5)(3,16,40,38,7,4)(10,26,33,29,44,14)(11,27,32,30,45,15)(12,25,31,28,43,13)(19,35,22,20,36,24)(21,34,23)$, $(1,16,37,7)(2,17,39,9)(3,18,38,8)(4,32)(5,31)(6,33)(10,27,30,44)(11,25,29,43)(12,26,28,45)(13,22)(14,23)(15,24)(19,34)(20,35)(21,36)(40,42)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2
$24$:  $S_4$
$36$:  $S_3^2$
$48$:  $S_4\times C_2$
$144$:  12T83
$6000$:  15T60

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 2

Degree 5: None

Degree 9: $S_3^2$

Degree 15: 15T60

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

66 x 66 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed