Properties

Label 16T3
16T3 1 2 1->2 6 1->6 10 1->10 15 1->15 5 2->5 9 2->9 16 2->16 3 4 3->4 3->5 11 3->11 3->16 4->6 12 4->12 4->15 5->6 14 5->14 13 6->13 7 8 7->8 7->10 7->12 7->15 8->9 8->11 8->16 9->10 9->14 10->13 11->12 11->14 12->13 13->14 15->16
Degree $16$
Order $16$
Cyclic no
Abelian yes
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2^4$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(16, 3);
 
Copy content sage:G = TransitiveGroup(16, 3)
 
Copy content oscar:G = transitive_group(16, 3)
 
Copy content gap:G := TransitiveGroup(16, 3);
 

Group invariants

Abstract group:  $C_2^4$
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:  $16=2^{4}$
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:  yes
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:  $1$
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 $t$:  $3$
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:  $(1,6)(2,5)(3,16)(4,15)(7,12)(8,11)(9,14)(10,13)$, $(1,10)(2,9)(3,11)(4,12)(5,14)(6,13)(7,15)(8,16)$, $(1,15)(2,16)(3,5)(4,6)(7,10)(8,9)(11,14)(12,13)$, $(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)$
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 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_2^2$ x 35

Degree 8: $C_2^3$ x 15

Low degree siblings

There are no siblings 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 $2^{8}$ $1$ $2$ $8$ $( 1, 6)( 2, 5)( 3,16)( 4,15)( 7,12)( 8,11)( 9,14)(10,13)$
2B $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,15)( 8,16)$
2C $2^{8}$ $1$ $2$ $8$ $( 1,15)( 2,16)( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,14)(12,13)$
2D $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2E $2^{8}$ $1$ $2$ $8$ $( 1,13)( 2,14)( 3, 8)( 4, 7)( 5, 9)( 6,10)(11,16)(12,15)$
2F $2^{8}$ $1$ $2$ $8$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$
2G $2^{8}$ $1$ $2$ $8$ $( 1, 7)( 2, 8)( 3,14)( 4,13)( 5,11)( 6,12)( 9,16)(10,15)$
2H $2^{8}$ $1$ $2$ $8$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,11)( 8,12)( 9,13)(10,14)$
2I $2^{8}$ $1$ $2$ $8$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$
2J $2^{8}$ $1$ $2$ $8$ $( 1,16)( 2,15)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)$
2K $2^{8}$ $1$ $2$ $8$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,15)(14,16)$
2L $2^{8}$ $1$ $2$ $8$ $( 1,14)( 2,13)( 3, 7)( 4, 8)( 5,10)( 6, 9)(11,15)(12,16)$
2M $2^{8}$ $1$ $2$ $8$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,14)( 8,13)( 9,12)(10,11)$
2N $2^{8}$ $1$ $2$ $8$ $( 1, 8)( 2, 7)( 3,13)( 4,14)( 5,12)( 6,11)( 9,15)(10,16)$
2O $2^{8}$ $1$ $2$ $8$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5, 7)( 6, 8)(13,16)(14,15)$

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 2H 2I 2J 2K 2L 2M 2N 2O
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A
Type
16.14.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16.14.1b R 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1
16.14.1c R 1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1
16.14.1d R 1 −1 −1 1 −1 1 −1 −1 1 1 −1 1 −1 1 1 −1
16.14.1e R 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1
16.14.1f R 1 −1 1 −1 −1 −1 1 −1 1 −1 1 1 1 −1 1 −1
16.14.1g R 1 −1 1 −1 1 −1 1 −1 −1 1 −1 1 −1 1 −1 1
16.14.1h R 1 −1 1 1 −1 −1 −1 1 1 −1 −1 −1 1 1 −1 1
16.14.1i R 1 −1 1 1 1 −1 −1 1 −1 1 1 −1 −1 −1 1 −1
16.14.1j R 1 1 −1 −1 −1 −1 −1 1 −1 1 1 1 1 1 −1 −1
16.14.1k R 1 1 −1 −1 1 −1 −1 1 1 −1 −1 1 −1 −1 1 1
16.14.1l R 1 1 −1 1 −1 −1 1 −1 −1 1 −1 −1 1 −1 1 1
16.14.1m R 1 1 −1 1 1 −1 1 −1 1 −1 1 −1 −1 1 −1 −1
16.14.1n R 1 1 1 −1 −1 1 −1 −1 −1 −1 1 −1 −1 1 1 1
16.14.1o R 1 1 1 −1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1
16.14.1p R 1 1 1 1 −1 1 1 1 −1 −1 −1 1 −1 −1 −1 −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