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Properties

Label 16T1704
16T1704 1 10 1->10 11 1->11 15 1->15 2 9 2->9 12 2->12 16 2->16 3 3->2 3->9 3->11 4 4->1 4->10 4->12 5 5->12 5->15 6 6->11 6->16 7 7->5 14 7->14 8 8->6 13 8->13 9->4 9->7 10->3 10->8 11->2 11->4 11->10 12->1 12->3 12->9 13->3 13->7 13->14 14->4 14->8 15->14 16->13
Degree $16$
Order $8192$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2^7.C_2\wr C_4$

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, 1704);
 
Copy content sage:G = TransitiveGroup(16, 1704)
 
Copy content oscar:G = transitive_group(16, 1704)
 
Copy content gap:G := TransitiveGroup(16, 1704);
 

Group invariants

Abstract group:  $C_2^7.C_2\wr C_4$
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:  $8192=2^{13}$
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:  $8$
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 $t$:  $1704$
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);
 
$\card{\Aut(F/K)}$:  $2$
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:  $(1,11,10,3,2,12,9,4)(5,15)(6,16)(7,14,8,13)$, $(1,15,14,4,10,8,6,11,2,16,13,3,9,7,5,12)$, $(1,10)(2,9)(3,11,4,12)(13,14)$
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
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 8, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
$32$:  $C_2^2 \wr C_2$, $C_2^3 : C_4 $ x 4, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
$64$:  $(C_4^2 : C_2):C_2$ x 2, $((C_8 : C_2):C_2):C_2$ x 2, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T76 x 2, 32T239
$128$:  16T219, 16T222, 16T227, 16T230, 16T234, 16T235, 16T302
$256$:  32T3918, 32T4019, 32T4050
$512$:  16T940
$1024$:  32T63498
$2048$:  16T1346
$4096$:  32T316637

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T1704 x 7, 16T1733 x 8, 32T399682 x 4, 32T399683 x 4, 32T399684 x 4, 32T399685 x 4, 32T399686 x 4, 32T399687 x 4, 32T399688 x 4, 32T399689 x 4, 32T399690 x 4, 32T399691 x 8, 32T399692 x 4, 32T399693 x 4, 32T399694 x 4, 32T399695 x 4, 32T399696 x 4, 32T400342 x 4, 32T400343 x 4, 32T400344 x 4, 32T400345 x 4, 32T400346 x 4, 32T400347 x 4, 32T400348 x 4, 32T400349 x 4, 32T400350 x 4, 32T400351 x 4, 32T400352 x 4, 32T400353 x 4, 32T400354 x 4, 32T400355 x 4, 32T406120 x 4, 32T406313 x 4, 32T406328 x 4, 32T433589 x 2, 32T433650 x 2, 32T433697 x 2, 32T433882 x 2, 32T433907 x 2, 32T433961 x 2, 32T542831 x 2, 32T543157 x 2, 32T545993 x 2, 32T546049 x 2, 32T644603 x 2, 32T644613 x 2, 32T644736 x 2, 32T644751 x 2, 32T668052 x 2, 32T668056 x 2, 32T684398 x 2, 32T684405 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Conjugacy classes not computed

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

83 x 83 character table

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