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Properties

Label 15T8
15T8 1 2 1->2 7 1->7 3 2->3 14 2->14 4 3->4 6 3->6 5 4->5 13 4->13 5->6 6->7 12 6->12 7->4 8 7->8 9 8->9 11 8->11 9->3 10 9->10 10->11 11->2 11->12 12->9 12->13 13->1 13->14 14->8 15 14->15 15->1
Degree $15$
Order $60$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $F_5\times C_3$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(15, 8);
 
Copy content sage:G = TransitiveGroup(15, 8)
 
Copy content oscar:G = transitive_group(15, 8)
 
Copy content gap:G := TransitiveGroup(15, 8);
 

Group invariants

Abstract group:  $F_5\times C_3$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $60=2^{2} \cdot 3 \cdot 5$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $15$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $8$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
CHM label:   $F(5)[x]3$
Parity:  $-1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(15).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(15), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(15), G));
 
Generators:  $(1,7,4,13)(2,14,8,11)(3,6,12,9)$, $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$
$20$:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $F_5$

Low degree siblings

30T7

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 $2^{6},1^{3}$ $5$ $2$ $6$ $( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$
3A1 $3^{5}$ $1$ $3$ $10$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
3A-1 $3^{5}$ $1$ $3$ $10$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
4A1 $4^{3},1^{3}$ $5$ $4$ $9$ $( 2, 8, 5,14)( 3,15, 9,12)( 4, 7,13,10)$
4A-1 $4^{3},1^{3}$ $5$ $4$ $9$ $( 2,14, 5, 8)( 3,12, 9,15)( 4,10,13, 7)$
5A $5^{3}$ $4$ $5$ $12$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
6A1 $6^{2},3$ $5$ $6$ $12$ $( 1, 6,11)( 2,10,12, 5, 7,15)( 3,14,13, 9, 8, 4)$
6A-1 $6^{2},3$ $5$ $6$ $12$ $( 1,11, 6)( 2,15, 7, 5,12,10)( 3, 4, 8, 9,13,14)$
12A1 $12,3$ $5$ $12$ $13$ $( 1,11, 6)( 2, 9,10, 8,12, 4, 5, 3, 7,14,15,13)$
12A-1 $12,3$ $5$ $12$ $13$ $( 1, 3, 2,10, 6, 8, 7,15,11,13,12, 5)( 4, 9,14)$
12A5 $12,3$ $5$ $12$ $13$ $( 1, 9, 8,10, 6,14,13,15,11, 4, 3, 5)( 2, 7,12)$
12A-5 $12,3$ $5$ $12$ $13$ $( 1, 5, 3, 4,11,15,13,14, 6,10, 8, 9)( 2,12, 7)$
15A1 $15$ $4$ $15$ $14$ $( 1,14,12,10, 8, 6, 4, 2,15,13,11, 9, 7, 5, 3)$
15A-1 $15$ $4$ $15$ $14$ $( 1,12, 8, 4,15,11, 7, 3,14,10, 6, 2,13, 9, 5)$

Malle's constant $a(G)$:     $1/6$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 4A1 4A-1 5A 6A1 6A-1 12A1 12A-1 12A5 12A-5 15A1 15A-1
Size 1 5 1 1 5 5 4 5 5 5 5 5 5 4 4
2 P 1A 1A 3A-1 3A1 2A 2A 5A 3A1 3A-1 6A1 6A-1 6A-1 6A1 15A-1 15A1
3 P 1A 2A 1A 1A 4A-1 4A1 5A 2A 2A 4A1 4A-1 4A1 4A-1 5A 5A
5 P 1A 2A 3A-1 3A1 4A1 4A-1 1A 6A-1 6A1 12A5 12A-5 12A1 12A-1 3A-1 3A1
Type
60.6.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
60.6.1b R 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1
60.6.1c1 C 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
60.6.1c2 C 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
60.6.1d1 C 1 −1 1 1 −i i 1 −1 −1 i −i i −i 1 1
60.6.1d2 C 1 −1 1 1 i −i 1 −1 −1 −i i −i i 1 1
60.6.1e1 C 1 1 ζ3−1 ζ3 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3
60.6.1e2 C 1 1 ζ3 ζ3−1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1
60.6.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 1 −ζ124 ζ122 −ζ125 ζ12 −ζ12 ζ125 −ζ122 ζ124
60.6.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 1 ζ122 −ζ124 ζ12 −ζ125 ζ125 −ζ12 ζ124 −ζ122
60.6.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 1 −ζ124 ζ122 ζ125 −ζ12 ζ12 −ζ125 −ζ122 ζ124
60.6.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 1 ζ122 −ζ124 −ζ12 ζ125 −ζ125 ζ12 ζ124 −ζ122
60.6.4a R 4 0 4 4 0 0 −1 0 0 0 0 0 0 −1 −1
60.6.4b1 C 4 0 4ζ3−1 4ζ3 0 0 −1 0 0 0 0 0 0 −ζ3−1 −ζ3
60.6.4b2 C 4 0 4ζ3 4ζ3−1 0 0 −1 0 0 0 0 0 0 −ζ3 −ζ3−1

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

Data not computed