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Properties

Label 15T20
15T20 1 5 1->5 9 1->9 2 7 2->7 15 2->15 3 6 3->6 14 3->14 4 4->5 4->15 11 5->11 6->2 12 7->12 8 8->4 8->9 10 9->10 10->3 13 11->13 12->6 12->13 13->8 14->1 15->7
Degree $15$
Order $360$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_6$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(15, 20);
 

Group invariants

Abstract group:  $A_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
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 $n$:  $15$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $20$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $A_{6}(15)$
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:  $(1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8)$, $(1,5)(2,7)(3,6)(4,15)(8,9)(12,13)$
Copy content magma:Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: None

Low degree siblings

6T15 x 2, 10T26, 15T20, 20T89, 30T88 x 2, 36T555, 40T304, 45T49

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{15}$ $1$ $1$ $0$ $()$
2A $2^{6},1^{3}$ $45$ $2$ $6$ $( 2,12)( 4, 6)( 5, 9)( 7,11)( 8,10)(14,15)$
3A $3^{4},1^{3}$ $40$ $3$ $8$ $( 1, 9,12)( 2, 7,10)( 3,14, 6)( 5,13, 8)$
3B $3^{5}$ $40$ $3$ $10$ $( 1, 9, 7)( 2,11, 6)( 3, 8, 4)( 5,15,10)(12,13,14)$
4A $4^{3},2,1$ $90$ $4$ $10$ $( 1,13)( 2, 6,12, 4)( 5, 7, 9,11)( 8,14,10,15)$
5A1 $5^{3}$ $72$ $5$ $12$ $( 1, 5, 7,14, 2)( 3,12,11,13, 6)( 4,10,15, 8, 9)$
5A2 $5^{3}$ $72$ $5$ $12$ $( 1, 7, 2, 5,14)( 3,11, 6,12,13)( 4,15, 9,10, 8)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A 3B 4A 5A1 5A2
Size 1 45 40 40 90 72 72
2 P 1A 1A 3A 3B 2A 5A2 5A1
3 P 1A 2A 1A 1A 4A 5A2 5A1
5 P 1A 2A 3A 3B 4A 1A 1A
Type
360.118.1a R 1 1 1 1 1 1 1
360.118.5a R 5 1 −1 2 −1 0 0
360.118.5b R 5 1 2 −1 −1 0 0
360.118.8a1 R 8 0 −1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52
360.118.8a2 R 8 0 −1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5
360.118.9a R 9 1 0 0 1 −1 −1
360.118.10a R 10 −2 1 1 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed