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Properties

Label 16T4
16T4 1 7 1->7 13 1->13 2 8 2->8 14 2->14 3 3->8 9 3->9 4 4->7 10 4->10 5 5->9 11 5->11 6 6->10 12 6->12 7->3 7->5 8->4 8->6 9->6 15 9->15 10->5 16 10->16 11->1 11->15 12->2 12->16 13->2 13->3 14->1 14->4 15->12 15->13 16->11 16->14
Degree $16$
Order $16$
Cyclic no
Abelian yes
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_4^2$

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, 4);
 
Copy content sage:G = TransitiveGroup(16, 4)
 
Copy content oscar:G = transitive_group(16, 4)
 
Copy content gap:G := TransitiveGroup(16, 4);
 

Group invariants

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 6, $C_2^2$

Degree 8: $C_4\times C_2$ x 3

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 $2^{8}$ $1$ $2$ $8$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,11)( 8,12)( 9,13)(10,14)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $1$ $2$ $8$ $( 1, 6)( 2, 5)( 3,16)( 4,15)( 7,12)( 8,11)( 9,14)(10,13)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 7, 5,11)( 2, 8, 6,12)( 3, 9,15,13)( 4,10,16,14)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1,11, 5, 7)( 2,12, 6, 8)( 3,13,15, 9)( 4,14,16,10)$
4B1 $4^{4}$ $1$ $4$ $12$ $( 1,13, 2,14)( 3, 8, 4, 7)( 5, 9, 6,10)(11,15,12,16)$
4B-1 $4^{4}$ $1$ $4$ $12$ $( 1,14, 2,13)( 3, 7, 4, 8)( 5,10, 6, 9)(11,16,12,15)$
4C1 $4^{4}$ $1$ $4$ $12$ $( 1, 8, 5,12)( 2, 7, 6,11)( 3,10,15,14)( 4, 9,16,13)$
4C-1 $4^{4}$ $1$ $4$ $12$ $( 1,12, 5, 8)( 2,11, 6, 7)( 3,14,15,10)( 4,13,16, 9)$
4D1 $4^{4}$ $1$ $4$ $12$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,15, 8,16)$
4D-1 $4^{4}$ $1$ $4$ $12$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,14, 6,13)( 7,16, 8,15)$
4E1 $4^{4}$ $1$ $4$ $12$ $( 1, 3, 6,16)( 2, 4, 5,15)( 7, 9,12,14)( 8,10,11,13)$
4E-1 $4^{4}$ $1$ $4$ $12$ $( 1,16, 6, 3)( 2,15, 5, 4)( 7,14,12, 9)( 8,13,11,10)$
4F1 $4^{4}$ $1$ $4$ $12$ $( 1,15, 6, 4)( 2,16, 5, 3)( 7,13,12,10)( 8,14,11, 9)$
4F-1 $4^{4}$ $1$ $4$ $12$ $( 1, 4, 6,15)( 2, 3, 5,16)( 7,10,12,13)( 8, 9,11,14)$

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 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1 4E1 4E-1 4F1 4F-1
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 1A 1A 2A 2A 2B 2B 2A 2A 2B 2B 2C 2C 2C 2C
Type
16.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16.2.1b R 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
16.2.1c R 1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 −1 −1
16.2.1d R 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1
16.2.1e1 C 1 −1 −1 1 −i i −i i i −i i −i −1 −1 1 1
16.2.1e2 C 1 −1 −1 1 i −i i −i −i i −i i −1 −1 1 1
16.2.1f1 C 1 −1 −1 1 −i i i −i i −i −i i 1 1 −1 −1
16.2.1f2 C 1 −1 −1 1 i −i −i i −i i i −i 1 1 −1 −1
16.2.1g1 C 1 −1 1 −1 −i i −1 −1 −i i 1 1 i −i −i i
16.2.1g2 C 1 −1 1 −1 i −i −1 −1 i −i 1 1 −i i i −i
16.2.1h1 C 1 −1 1 −1 −i i 1 1 −i i −1 −1 −i i i −i
16.2.1h2 C 1 −1 1 −1 i −i 1 1 i −i −1 −1 i −i −i i
16.2.1i1 C 1 1 −1 −1 −1 −1 −i i 1 1 −i i i −i i −i
16.2.1i2 C 1 1 −1 −1 −1 −1 i −i 1 1 i −i −i i −i i
16.2.1j1 C 1 1 −1 −1 1 1 −i i −1 −1 −i i −i i −i i
16.2.1j2 C 1 1 −1 −1 1 1 i −i −1 −1 i −i i −i i −i

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