Properties

Label 16T40
16T40 1 11 1->11 15 1->15 2 12 2->12 16 2->16 3 9 3->9 13 3->13 4 10 4->10 14 4->14 5 5->12 5->15 6 6->11 6->16 7 7->10 7->13 8 8->9 8->14 9->1 9->5 10->2 10->6 11->3 11->7 12->4 12->8 13->2 13->5 14->1 14->6 15->4 15->7 16->3 16->8
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_4.D_4$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(16, 40);
 
Copy content sage:G = TransitiveGroup(16, 40)
 
Copy content oscar:G = transitive_group(16, 40)
 
Copy content gap:G := TransitiveGroup(16, 40);
 

Group invariants

Abstract group:  $C_4.D_4$
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:  $32=2^{5}$
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:  $3$
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 $t$:  $40$
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);
 
$\card{\Aut(F/K)}$:  $8$
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:  $(1,11,3,13,2,12,4,14)(5,15,7,10,6,16,8,9)$, $(1,15,4,10,2,16,3,9)(5,12,8,14,6,11,7,13)$
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, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $C_4\times C_2$

Low degree siblings

32T25

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

Malle's constant $a(G)$:     $1/4$

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 4A 4B 4C 4D 8A1 8A-1 8B1 8B-1
Size 1 1 2 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 2A 2A 2A 2A 4A 4A 4B 4B
Type
32.8.1a R 1 1 1 1 1 1 1 1 1 1 1
32.8.1b R 1 1 1 1 1 −1 −1 −1 −1 1 1
32.8.1c R 1 1 1 1 1 −1 −1 1 1 −1 −1
32.8.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1
32.8.1e1 C 1 1 1 −1 −1 −1 1 −i i i −i
32.8.1e2 C 1 1 1 −1 −1 −1 1 i −i −i i
32.8.1f1 C 1 1 1 −1 −1 1 −1 −i i −i i
32.8.1f2 C 1 1 1 −1 −1 1 −1 i −i i −i
32.8.2a R 2 2 −2 −2 2 0 0 0 0 0 0
32.8.2b R 2 2 −2 2 −2 0 0 0 0 0 0
32.8.4a S 4 −4 0 0 0 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