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Properties

Label 16T9
16T9 1 3 1->3 11 1->11 13 1->13 2 4 2->4 12 2->12 14 2->14 8 3->8 10 3->10 7 4->7 9 4->9 5 5->8 5->10 15 5->15 6 6->7 6->9 16 6->16 7->3 7->11 8->4 8->12 9->5 9->13 10->6 10->14 11->15 11->16 12->15 12->16 13->2 14->1 15->13 16->14
Degree $16$
Order $16$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $D_4\times C_2$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(16, 9);
 
Copy content sage:G = TransitiveGroup(16, 9)
 
Copy content oscar:G = transitive_group(16, 9)
 
Copy content gap:G := TransitiveGroup(16, 9);
 

Group invariants

Abstract group:  $D_4\times C_2$
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:  $16=2^{4}$
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:  $2$
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 $n$:  $16$
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 $t$:  $9$
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);
 
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);
 
$\card{\Aut(F/K)}$:  $16$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(16).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(16), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(16), G));
 
Generators:  $(1,3)(2,4)(5,15)(6,16)(7,11)(8,12)(9,13)(10,14)$, $(1,11,16,14)(2,12,15,13)(3,10,6,7)(4,9,5,8)$, $(1,13)(2,14)(3,8)(4,7)(5,10)(6,9)(11,15)(12,16)$
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 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4

Low degree siblings

8T9 x 4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{16}$ $1$ $1$ $0$ $()$
2A $2^{8}$ $1$ $2$ $8$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,12)( 8,11)( 9,14)(10,13)$
2C $2^{8}$ $1$ $2$ $8$ $( 1,16)( 2,15)( 3, 6)( 4, 5)( 7,10)( 8, 9)(11,14)(12,13)$
2D $2^{8}$ $2$ $2$ $8$ $( 1,15)( 2,16)( 3, 5)( 4, 6)( 7, 8)( 9,10)(11,12)(13,14)$
2E $2^{8}$ $2$ $2$ $8$ $( 1, 6)( 2, 5)( 3,16)( 4,15)( 7,14)( 8,13)( 9,12)(10,11)$
2F $2^{8}$ $2$ $2$ $8$ $( 1,13)( 2,14)( 3, 8)( 4, 7)( 5,10)( 6, 9)(11,15)(12,16)$
2G $2^{8}$ $2$ $2$ $8$ $( 1, 7)( 2, 8)( 3,14)( 4,13)( 5,12)( 6,11)( 9,15)(10,16)$
4A $4^{4}$ $2$ $4$ $12$ $( 1,11,16,14)( 2,12,15,13)( 3,10, 6, 7)( 4, 9, 5, 8)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 9,16, 8)( 2,10,15, 7)( 3,12, 6,13)( 4,11, 5,14)$

Malle's constant $a(G)$:     $1/8$

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 2D 2E 2F 2G 4A 4B
Size 1 1 1 1 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2C 2C
Type
16.11.1a R 1 1 1 1 1 1 1 1 1 1
16.11.1b R 1 −1 −1 1 −1 1 −1 1 1 −1
16.11.1c R 1 −1 −1 1 −1 1 1 −1 −1 1
16.11.1d R 1 −1 −1 1 1 −1 −1 1 −1 1
16.11.1e R 1 −1 −1 1 1 −1 1 −1 1 −1
16.11.1f R 1 1 1 1 −1 −1 −1 −1 1 1
16.11.1g R 1 1 1 1 −1 −1 1 1 −1 −1
16.11.1h R 1 1 1 1 1 1 −1 −1 −1 −1
16.11.2a R 2 −2 2 −2 0 0 0 0 0 0
16.11.2b R 2 2 −2 −2 0 0 0 0 0 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

Data not computed