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Properties

Label 16T6
16T6 1 3 1->3 9 1->9 2 4 2->4 10 2->10 3->4 5 3->5 6 4->6 8 5->8 13 5->13 7 6->7 14 6->14 7->8 7->9 8->10 11 9->11 12 10->12 11->12 11->13 12->14 16 13->16 15 14->15 15->1 15->16 16->2
Degree $16$
Order $16$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_8: C_2$

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, 6);
 
Copy content sage:G = TransitiveGroup(16, 6)
 
Copy content oscar:G = transitive_group(16, 6)
 
Copy content gap:G := TransitiveGroup(16, 6);
 

Group invariants

Abstract group:  $C_8: C_2$
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 $t$:  $6$
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:  $(1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16)$, $(1,3,5,8,10,12,14,15)(2,4,6,7,9,11,13,16)$
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 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$, $C_8:C_2$

Low degree siblings

8T7

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 $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
2B $2^{8}$ $2$ $2$ $8$ $( 1, 9)( 2,10)( 3, 4)( 5,13)( 6,14)( 7, 8)(11,12)(15,16)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 5,10,14)( 2, 6, 9,13)( 3, 8,12,15)( 4, 7,11,16)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1,14,10, 5)( 2,13, 9, 6)( 3,15,12, 8)( 4,16,11, 7)$
4B $4^{4}$ $2$ $4$ $12$ $( 1,13,10, 6)( 2,14, 9, 5)( 3, 7,12,16)( 4, 8,11,15)$
8A1 $8^{2}$ $2$ $8$ $14$ $( 1, 3, 5, 8,10,12,14,15)( 2, 4, 6, 7, 9,11,13,16)$
8A-1 $8^{2}$ $2$ $8$ $14$ $( 1, 8,14, 3,10,15, 5,12)( 2, 7,13, 4, 9,16, 6,11)$
8B1 $8^{2}$ $2$ $8$ $14$ $( 1,11,14, 7,10, 4, 5,16)( 2,12,13, 8, 9, 3, 6,15)$
8B-1 $8^{2}$ $2$ $8$ $14$ $( 1,16, 5, 4,10, 7,14,11)( 2,15, 6, 3, 9, 8,13,12)$

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 4A1 4A-1 4B 8A1 8A-1 8B1 8B-1
Size 1 1 2 1 1 2 2 2 2 2
2 P 1A 1A 1A 2A 2A 2A 4A1 4A-1 4A-1 4A1
Type
16.6.1a R 1 1 1 1 1 1 1 1 1 1
16.6.1b R 1 1 −1 1 1 −1 −1 −1 1 1
16.6.1c R 1 1 −1 1 1 −1 1 1 −1 −1
16.6.1d R 1 1 1 1 1 1 −1 −1 −1 −1
16.6.1e1 C 1 1 −1 −1 −1 1 −i i i −i
16.6.1e2 C 1 1 −1 −1 −1 1 i −i −i i
16.6.1f1 C 1 1 1 −1 −1 −1 −i i −i i
16.6.1f2 C 1 1 1 −1 −1 −1 i −i i −i
16.6.2a1 C 2 −2 0 −2i 2i 0 0 0 0 0
16.6.2a2 C 2 −2 0 2i −2i 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