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Properties

Label 15T4
15T4 1 2 1->2 11 1->11 3 2->3 7 2->7 4 3->4 5 4->5 14 4->14 6 5->6 10 5->10 6->7 8 7->8 9 8->9 13 8->13 9->10 10->11 12 11->12 12->13 13->14 15 14->15 15->1
Degree $15$
Order $30$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3 \times C_5$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(15, 4);
 
Copy content sage:G = TransitiveGroup(15, 4)
 
Copy content oscar:G = transitive_group(15, 4)
 

Group invariants

Abstract group:  $S_3 \times C_5$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $30=2 \cdot 3 \cdot 5$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar: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)
 
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)
 
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)
 
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
 

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)
 
Transitive number $t$:  $4$
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]
 
CHM label:   $5[x]S(3)$
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)
 
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)
 
$\card{\Aut(F/K)}$:  $5$
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])
 
Generators:  $(1,11)(2,7)(4,14)(5,10)(8,13)$, $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $C_5$

Low degree siblings

30T2

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 $2^{5},1^{5}$ $3$ $2$ $5$ $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$
3A $3^{5}$ $2$ $3$ $10$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
5A1 $5^{3}$ $1$ $5$ $12$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
5A-1 $5^{3}$ $1$ $5$ $12$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
5A2 $5^{3}$ $1$ $5$ $12$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
5A-2 $5^{3}$ $1$ $5$ $12$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
10A1 $10,5$ $3$ $10$ $13$ $( 1, 7,13, 4,10)( 2, 3,14,15,11,12, 8, 9, 5, 6)$
10A-1 $10,5$ $3$ $10$ $13$ $( 1,15, 4, 3, 7, 6,10, 9,13,12)( 2,11, 5,14, 8)$
10A3 $10,5$ $3$ $10$ $13$ $( 1, 4, 7,10,13)( 2,15, 8, 6,14,12, 5, 3,11, 9)$
10A-3 $10,5$ $3$ $10$ $13$ $( 1, 8,10, 2, 4,11,13, 5, 7,14)( 3,15,12, 9, 6)$
15A1 $15$ $2$ $15$ $14$ $( 1, 8,15, 7,14, 6,13, 5,12, 4,11, 3,10, 2, 9)$
15A-1 $15$ $2$ $15$ $14$ $( 1,14,12,10, 8, 6, 4, 2,15,13,11, 9, 7, 5, 3)$
15A2 $15$ $2$ $15$ $14$ $( 1,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
15A-2 $15$ $2$ $15$ $14$ $( 1,12, 8, 4,15,11, 7, 3,14,10, 6, 2,13, 9, 5)$

Malle's constant $a(G)$:     $1/5$

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

Character table

1A 2A 3A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 15A1 15A-1 15A2 15A-2
Size 1 3 2 1 1 1 1 3 3 3 3 2 2 2 2
2 P 1A 1A 3A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 15A2 15A-2 15A-1 15A1
3 P 1A 2A 1A 5A-2 5A2 5A1 5A-1 10A3 10A-3 10A-1 10A1 5A-2 5A2 5A1 5A-1
5 P 1A 2A 3A 1A 1A 1A 1A 2A 2A 2A 2A 3A 3A 3A 3A
Type
30.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
30.1.1b R 1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1
30.1.1c1 C 1 1 1 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−1 ζ52 ζ5 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1
30.1.1c2 C 1 1 1 ζ52 ζ5 ζ5−1 ζ5−2 ζ5 ζ5−2 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−1 ζ5
30.1.1c3 C 1 1 1 ζ5−1 ζ52 ζ5−2 ζ5 ζ52 ζ5 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52
30.1.1c4 C 1 1 1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5−2 ζ5−1 ζ52 ζ5 ζ5−1 ζ5 ζ52 ζ5−2
30.1.1d1 C 1 −1 1 ζ5−2 ζ5−1 ζ5 ζ52 −ζ5−1 −ζ52 −ζ5 −ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1
30.1.1d2 C 1 −1 1 ζ52 ζ5 ζ5−1 ζ5−2 −ζ5 −ζ5−2 −ζ5−1 −ζ52 ζ5−2 ζ52 ζ5−1 ζ5
30.1.1d3 C 1 −1 1 ζ5−1 ζ52 ζ5−2 ζ5 −ζ52 −ζ5 −ζ5−2 −ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52
30.1.1d4 C 1 −1 1 ζ5 ζ5−2 ζ52 ζ5−1 −ζ5−2 −ζ5−1 −ζ52 −ζ5 ζ5−1 ζ5 ζ52 ζ5−2
30.1.2a R 2 0 −1 2 2 2 2 0 0 0 0 −1 −1 −1 −1
30.1.2b1 C 2 0 −1 2ζ5−2 2ζ5−1 2ζ5 2ζ52 0 0 0 0 −ζ52 −ζ5−2 −ζ5 −ζ5−1
30.1.2b2 C 2 0 −1 2ζ52 2ζ5 2ζ5−1 2ζ5−2 0 0 0 0 −ζ5−2 −ζ52 −ζ5−1 −ζ5
30.1.2b3 C 2 0 −1 2ζ5−1 2ζ52 2ζ5−2 2ζ5 0 0 0 0 −ζ5 −ζ5−1 −ζ5−2 −ζ52
30.1.2b4 C 2 0 −1 2ζ5 2ζ5−2 2ζ52 2ζ5−1 0 0 0 0 −ζ5−1 −ζ5 −ζ52 −ζ5−2

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

Regular extensions

Data not computed