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Properties

Label 15T11
15T11 1 2 1->2 7 1->7 11 1->11 3 2->3 2->7 14 2->14 4 3->4 6 3->6 5 4->5 13 4->13 4->14 5->6 10 5->10 6->7 12 6->12 7->4 8 7->8 9 8->9 8->11 8->13 9->3 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 $120$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $F_5 \times S_3$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(15, 11);
 
Copy content sage:G = TransitiveGroup(15, 11)
 
Copy content oscar:G = transitive_group(15, 11)
 
Copy content gap:G := TransitiveGroup(15, 11);
 

Group invariants

Abstract group:  $F_5 \times S_3$
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:  $120=2^{3} \cdot 3 \cdot 5$
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 $t$:  $11$
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:   $F(5)[x]S(3)$
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)
 
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);
 
$\card{\Aut(F/K)}$:  $1$
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:  $(1,11)(2,7)(4,14)(5,10)(8,13)$, $(1,7,4,13)(2,14,8,11)(3,6,12,9)$, $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)$
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)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$20$:  $F_5$
$24$:  $S_3 \times C_4$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $F_5$

Low degree siblings

30T23, 30T24, 30T32

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 $2^{5},1^{5}$ $3$ $2$ $5$ $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$
2B $2^{6},1^{3}$ $5$ $2$ $6$ $( 1,10)( 2,14)( 4, 7)( 5,11)( 6,15)( 9,12)$
2C $2^{7},1$ $15$ $2$ $7$ $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,15)(13,14)$
3A $3^{5}$ $2$ $3$ $10$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
4A1 $4^{3},1^{3}$ $5$ $4$ $9$ $( 1, 4,10, 7)( 2,11,14, 5)( 6, 9,15,12)$
4A-1 $4^{3},1^{3}$ $5$ $4$ $9$ $( 1, 7,10, 4)( 2, 5,14,11)( 6,12,15, 9)$
4B1 $4^{3},2,1$ $15$ $4$ $10$ $( 1, 3, 7,15)( 2, 5,11, 8)( 4, 9)( 6,13,12,10)$
4B-1 $4^{3},2,1$ $15$ $4$ $10$ $( 1,15, 7, 3)( 2, 8,11, 5)( 4, 9)( 6,10,12,13)$
5A $5^{3}$ $4$ $5$ $12$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
6A $6^{2},3$ $10$ $6$ $12$ $( 1, 5, 6,10,11,15)( 2, 9, 7,14,12, 4)( 3,13, 8)$
10A $10,5$ $12$ $10$ $13$ $( 1,10, 4,13, 7)( 2, 6, 5, 9, 8,12,11,15,14, 3)$
12A1 $12,3$ $10$ $12$ $13$ $( 1,12, 5, 4, 6, 2,10, 9,11, 7,15,14)( 3, 8,13)$
12A-1 $12,3$ $10$ $12$ $13$ $( 1, 6,11)( 2,13,15,14, 7, 3, 5, 4,12, 8,10, 9)$
15A $15$ $8$ $15$ $14$ $( 1,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

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)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 5A 6A 10A 12A1 12A-1 15A
Size 1 3 5 15 2 5 5 15 15 4 10 12 10 10 8
2 P 1A 1A 1A 1A 3A 2B 2B 2B 2B 5A 3A 5A 6A 6A 15A
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 5A 2B 10A 4A1 4A-1 5A
5 P 1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 1A 6A 2A 12A1 12A-1 3A
Type
120.36.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
120.36.1b R 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1
120.36.1c R 1 −1 1 −1 1 1 1 −1 −1 1 1 −1 1 1 1
120.36.1d R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1
120.36.1e1 C 1 −1 −1 1 1 −i i i −i 1 −1 −1 i −i 1
120.36.1e2 C 1 −1 −1 1 1 i −i −i i 1 −1 −1 −i i 1
120.36.1f1 C 1 1 −1 −1 1 −i i −i i 1 −1 1 i −i 1
120.36.1f2 C 1 1 −1 −1 1 i −i i −i 1 −1 1 −i i 1
120.36.2a R 2 0 2 0 −1 2 2 0 0 2 −1 0 −1 −1 −1
120.36.2b R 2 0 2 0 −1 −2 −2 0 0 2 −1 0 1 1 −1
120.36.2c1 C 2 0 −2 0 −1 −2i 2i 0 0 2 1 0 −i i −1
120.36.2c2 C 2 0 −2 0 −1 2i −2i 0 0 2 1 0 i −i −1
120.36.4a R 4 4 0 0 4 0 0 0 0 −1 0 −1 0 0 −1
120.36.4b R 4 −4 0 0 4 0 0 0 0 −1 0 1 0 0 −1
120.36.8a R 8 0 0 0 −4 0 0 0 0 −2 0 0 0 0 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