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Properties

Label 16T97
16T97 1 6 1->6 10 1->10 16 1->16 1->16 2 5 2->5 9 2->9 15 2->15 2->15 3 3->5 11 3->11 3->16 4 4->6 12 4->12 4->15 5->1 14 5->14 6->2 13 6->13 7 7->10 7->10 7->11 7->15 8 8->9 8->9 8->12 8->16 9->1 9->13 10->2 10->14 11->4 11->8 11->14 12->3 12->7 12->13 13->5 13->10 14->6 14->9 15->3 15->8 16->4 16->7
Degree $16$
Order $64$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2^3:Q_8$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(16, 97);
 
Copy content sage:G = TransitiveGroup(16, 97)
 
Copy content oscar:G = transitive_group(16, 97)
 
Copy content gap:G := TransitiveGroup(16, 97);
 

Group invariants

Abstract group:  $C_2^3:Q_8$
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:  $64=2^{6}$
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 $t$:  $97$
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);
 
$\card{\Aut(F/K)}$:  $4$
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:  $(1,16)(2,15)(3,5)(4,6)$, $(1,10,2,9)(3,11,4,12)(5,14,6,13)(7,15,8,16)$, $(7,10)(8,9)(11,14)(12,13)$, $(1,6,2,5)(3,16,4,15)(7,11,8,12)(9,13,10,14)$, $(1,16)(2,15)(7,10)(8,9)$
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, $Q_8$ x 4
$16$:  $C_2^4$, $Q_8\times C_2$ x 6
$32$:  $Q_8:C_2^2$ x 2, 32T40

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8$, $Q_8:C_2^2$ x 2

Low degree siblings

16T97 x 3, 32T98 x 6

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 $2^{8}$ $1$ $2$ $8$ $( 1,16)( 2,15)( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,14)(12,13)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $1$ $2$ $8$ $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)$
2D $2^{4},1^{8}$ $2$ $2$ $4$ $( 1,16)( 2,15)(11,14)(12,13)$
2E $2^{8}$ $2$ $2$ $8$ $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7, 8)( 9,10)(11,12)(13,14)$
2F $2^{8}$ $2$ $2$ $8$ $( 1,15)( 2,16)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,13)(12,14)$
2G $2^{4},1^{8}$ $2$ $2$ $4$ $( 1,16)( 2,15)( 3, 5)( 4, 6)$
2H $2^{8}$ $2$ $2$ $8$ $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,10)(11,13)(12,14)(15,16)$
2I $2^{4},1^{8}$ $2$ $2$ $4$ $( 3, 5)( 4, 6)(11,14)(12,13)$
4A $4^{4}$ $4$ $4$ $12$ $( 1,11, 2,12)( 3, 9, 4,10)( 5, 8, 6, 7)(13,16,14,15)$
4B $4^{4}$ $4$ $4$ $12$ $( 1, 3,15, 6)( 2, 4,16, 5)( 7,12, 9,14)( 8,11,10,13)$
4C $4^{4}$ $4$ $4$ $12$ $( 1,14,15,12)( 2,13,16,11)( 3, 8, 6,10)( 4, 7, 5, 9)$
4D $4^{4}$ $4$ $4$ $12$ $( 1, 9,15, 7)( 2,10,16, 8)( 3,13, 6,11)( 4,14, 5,12)$
4E $4^{4}$ $4$ $4$ $12$ $( 1, 7, 2, 8)( 3,11, 4,12)( 5,14, 6,13)( 9,16,10,15)$
4F $4^{4}$ $4$ $4$ $12$ $( 1,14, 2,13)( 3, 9, 4,10)( 5, 8, 6, 7)(11,15,12,16)$
4G $4^{4}$ $4$ $4$ $12$ $( 1, 5, 2, 6)( 3,15, 4,16)( 7,12, 8,11)( 9,14,10,13)$
4H $4^{4}$ $4$ $4$ $12$ $( 1,11,15,13)( 2,12,16,14)( 3, 8, 6,10)( 4, 7, 5, 9)$
4I $4^{4}$ $4$ $4$ $12$ $( 1, 5,15, 4)( 2, 6,16, 3)( 7,12, 9,14)( 8,11,10,13)$
4J $4^{4}$ $4$ $4$ $12$ $( 1, 8, 2, 7)( 3,13, 4,14)( 5,12, 6,11)( 9,15,10,16)$
4K $4^{4}$ $4$ $4$ $12$ $( 1,10,15, 8)( 2, 9,16, 7)( 3,11, 6,13)( 4,12, 5,14)$
4L $4^{4}$ $4$ $4$ $12$ $( 1, 3, 2, 4)( 5,15, 6,16)( 7,12, 8,11)( 9,14,10,13)$

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 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L
Size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2B 2C 2C 2C 2B 2B 2B 2C 2C 2B 2C 2B
Type
64.224.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.224.1b R 1 1 1 1 −1 −1 1 −1 −1 1 −1 −1 1 1 1 1 −1 1 −1 −1 1 −1
64.224.1c R 1 1 1 1 −1 −1 1 −1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 1 −1 1
64.224.1d R 1 1 1 1 −1 −1 1 −1 −1 1 1 −1 −1 −1 1 −1 1 1 1 1 −1 −1
64.224.1e R 1 1 1 1 −1 −1 1 −1 −1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1
64.224.1f R 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1 −1 −1 1
64.224.1g R 1 1 1 1 −1 1 −1 −1 1 −1 −1 1 1 −1 −1 −1 1 1 −1 1 1 −1
64.224.1h R 1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 −1 −1 −1 1 1 1
64.224.1i R 1 1 1 1 −1 1 −1 −1 1 −1 1 1 −1 1 −1 1 −1 1 1 −1 −1 −1
64.224.1j R 1 1 1 1 1 −1 −1 1 −1 −1 −1 −1 −1 1 −1 1 1 1 −1 1 −1 1
64.224.1k R 1 1 1 1 1 −1 −1 1 −1 −1 −1 1 −1 −1 1 1 1 −1 1 −1 1 −1
64.224.1l R 1 1 1 1 1 −1 −1 1 −1 −1 1 −1 1 −1 −1 −1 −1 1 1 −1 1 1
64.224.1m R 1 1 1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1
64.224.1n R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 1 −1 −1 −1 −1 1 1 1 −1
64.224.1o R 1 1 1 1 1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 1 −1 −1 −1 1
64.224.1p R 1 1 1 1 1 1 1 1 1 1 1 −1 1 −1 −1 1 1 −1 −1 −1 −1 −1
64.224.2a S 2 −2 2 −2 −2 −2 2 2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
64.224.2b S 2 −2 2 −2 −2 2 −2 2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
64.224.2c S 2 −2 2 −2 2 −2 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
64.224.2d S 2 −2 2 −2 2 2 2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
64.224.4a R 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
64.224.4b R 4 4 −4 −4 0 0 0 0 0 0 0 0 0 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