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Properties

Label 15T17
15T17 1 2 1->2 4 1->4 6 1->6 13 1->13 2->4 5 2->5 7 2->7 8 2->8 3 3->6 3->8 12 3->12 4->1 4->8 9 4->9 5->8 10 5->10 5->10 6->9 11 6->11 6->12 7->4 7->12 7->13 14 7->14 8->1 8->11 8->13 9->3 9->14 10->7 15 10->15 11->1 11->7 11->14 11->14 12->2 12->9 13->3 13->10 13->11 14->2 14->4 14->13 15->5
Degree $15$
Order $300$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $(C_5^2 : C_3):C_4$

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, 17);
 
Copy content sage:G = TransitiveGroup(15, 17)
 
Copy content oscar:G = transitive_group(15, 17)
 
Copy content gap:G := TransitiveGroup(15, 17);
 

Group invariants

Abstract group:  $(C_5^2 : C_3):C_4$
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:  $300=2^{2} \cdot 3 \cdot 5^{2}$
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 $t$:  $17$
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:   $1/2[5^{2}:4]S(3)$
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:  $(1,2,4,8)(3,6,12,9)(5,10)(7,14,13,11)$, $(1,4)(2,8)(3,12)(6,9)(7,13)(11,14)$, $(1,13,10,7,4)(2,5,8,11,14)$, $(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)$
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)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$12$:  $C_3 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

15T17, 25T28, 30T71 x 2

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 $2^{6},1^{3}$ $25$ $2$ $6$ $( 1, 7)( 2,11)( 3, 6)( 5, 8)( 9,15)(10,13)$
3A $3^{5}$ $50$ $3$ $10$ $( 1,11,15)( 2, 6, 7)( 3, 4,14)( 5, 9,10)( 8,12,13)$
4A1 $4^{3},2,1$ $75$ $4$ $10$ $( 1, 6, 7, 3)( 2, 5,11, 8)( 4,12)( 9,13,15,10)$
4A-1 $4^{3},2,1$ $75$ $4$ $10$ $( 1, 3, 7, 6)( 2, 8,11, 5)( 4,12)( 9,10,15,13)$
5A $5^{2},1^{5}$ $12$ $5$ $8$ $( 1,13,10, 7, 4)( 2, 5, 8,11,14)$
5B $5^{3}$ $12$ $5$ $12$ $( 1,10, 4,13, 7)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
6A $6^{2},3$ $50$ $6$ $12$ $( 1,12,11,13,15, 8)( 2, 7, 6)( 3, 5, 4, 9,14,10)$

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 3A 4A1 4A-1 5A 5B 6A
Size 1 25 50 75 75 12 12 50
2 P 1A 1A 3A 2A 2A 5A 5B 3A
3 P 1A 2A 1A 4A-1 4A1 5A 5B 2A
5 P 1A 2A 3A 4A1 4A-1 1A 1A 6A
Type
300.23.1a R 1 1 1 1 1 1 1 1
300.23.1b R 1 1 1 −1 −1 1 1 1
300.23.1c1 C 1 −1 1 −i i 1 1 −1
300.23.1c2 C 1 −1 1 i −i 1 1 −1
300.23.2a R 2 2 −1 0 0 2 2 −1
300.23.2b S 2 −2 −1 0 0 2 2 1
300.23.12a R 12 0 0 0 0 −3 2 0
300.23.12b R 12 0 0 0 0 2 −3 0

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

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