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Properties

Label 16T247
16T247 1 4 1->4 5 1->5 7 1->7 2 3 2->3 6 2->6 8 2->8 3->5 13 3->13 15 3->15 4->6 14 4->14 16 4->16 11 5->11 5->16 12 6->12 6->15 7->4 9 7->9 7->14 8->3 10 8->10 8->13 9->6 9->12 9->12 10->5 10->11 10->11 11->13 11->16 12->14 12->15 13->2 13->8 14->1 14->7 15->1 15->9 16->2 16->10
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2^4.D_4$

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, 247);
 
Copy content sage:G = TransitiveGroup(16, 247)
 
Copy content oscar:G = transitive_group(16, 247)
 
Copy content gap:G := TransitiveGroup(16, 247);
 

Group invariants

Abstract group:  $C_2^4.D_4$
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:  $128=2^{7}$
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 $t$:  $247$
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:  $(1,5)(2,6)(3,15)(4,16)(7,14)(8,13)(9,12)(10,11)$, $(1,7,4,14)(2,8,3,13)(5,11,16,10)(6,12,15,9)$, $(1,4,6,15)(2,3,5,16)(7,9,12,14)(8,10,11,13)$
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 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, $C_4\times C_2^2$
$32$:  $C_2^2 \wr C_2$, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 32T239

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_2^2:C_4$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2

Low degree siblings

16T208 x 4, 16T210 x 4, 16T217 x 8, 16T247 x 7, 32T451 x 2, 32T452, 32T453 x 4, 32T454 x 8, 32T455 x 4, 32T459 x 8, 32T460, 32T478 x 4, 32T479 x 2, 32T564 x 2, 32T565 x 2, 32T1567 x 2, 32T1705

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 $2^{8}$ $1$ $2$ $8$ $( 1, 6)( 2, 5)( 3,16)( 4,15)( 7,12)( 8,11)( 9,14)(10,13)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,11)( 8,12)( 9,13)(10,14)$
2C $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2D $2^{8}$ $2$ $2$ $8$ $( 1,16)( 2,15)( 3, 6)( 4, 5)( 7,13)( 8,14)( 9,11)(10,12)$
2E $2^{8}$ $2$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7,11)( 8,12)( 9,13)(10,14)(15,16)$
2F $2^{8}$ $2$ $2$ $8$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$
2G $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 6)( 2, 5)( 3,16)( 4,15)$
2H $2^{8}$ $2$ $2$ $8$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7, 9)( 8,10)(11,13)(12,14)$
2I $2^{8}$ $2$ $2$ $8$ $( 1,15)( 2,16)( 3, 5)( 4, 6)( 7, 9)( 8,10)(11,13)(12,14)$
2J $2^{4},1^{8}$ $4$ $2$ $4$ $( 1,16)( 2,15)( 3, 6)( 4, 5)$
2K $2^{8}$ $4$ $2$ $8$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,11)( 8,12)( 9,13)(10,14)$
2L $2^{4},1^{8}$ $4$ $2$ $4$ $( 3,16)( 4,15)( 7,12)( 8,11)$
2M $2^{8}$ $4$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 4)( 7, 8)( 9,13)(10,14)(11,12)(15,16)$
2N $2^{8}$ $4$ $2$ $8$ $( 1, 6)( 2, 5)( 3,16)( 4,15)( 7,13)( 8,14)( 9,11)(10,12)$
2O $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9)( 8,10)(11,13)(12,14)(15,16)$
4A $4^{4}$ $4$ $4$ $12$ $( 1, 3, 6,16)( 2, 4, 5,15)( 7,13,12,10)( 8,14,11, 9)$
4B $4^{4}$ $4$ $4$ $12$ $( 1,15, 6, 4)( 2,16, 5, 3)( 7, 9,12,14)( 8,10,11,13)$
4C1 $4^{4}$ $4$ $4$ $12$ $( 1,10, 5,14)( 2, 9, 6,13)( 3, 7,15,11)( 4, 8,16,12)$
4C-1 $4^{4}$ $4$ $4$ $12$ $( 1,14, 5,10)( 2,13, 6, 9)( 3,11,15, 7)( 4,12,16, 8)$
4D1 $4^{4}$ $4$ $4$ $12$ $( 1,10, 5,14)( 2, 9, 6,13)( 3,12,15, 8)( 4,11,16, 7)$
4D-1 $4^{4}$ $4$ $4$ $12$ $( 1,14, 5,10)( 2,13, 6, 9)( 3, 8,15,12)( 4, 7,16,11)$
4E1 $4^{4}$ $4$ $4$ $12$ $( 1, 7, 2, 8)( 3,13, 4,14)( 5,11, 6,12)( 9,16,10,15)$
4E-1 $4^{4}$ $4$ $4$ $12$ $( 1,11, 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,16,14,15)$
4F1 $4^{4}$ $4$ $4$ $12$ $( 1,10, 2, 9)( 3, 7, 4, 8)( 5,14, 6,13)(11,16,12,15)$
4F-1 $4^{4}$ $4$ $4$ $12$ $( 1,14, 2,13)( 3,11, 4,12)( 5,10, 6, 9)( 7,16, 8,15)$
4G $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 3, 6,16)( 2, 4, 5,15)( 7,12)( 8,11)$
4H $4^{2},2^{4}$ $8$ $4$ $10$ $( 1,15, 6, 4)( 2,16, 5, 3)( 7, 8)( 9,13)(10,14)(11,12)$
4I1 $4^{4}$ $8$ $4$ $12$ $( 1,10, 4, 8)( 2, 9, 3, 7)( 5,14,16,12)( 6,13,15,11)$
4I-1 $4^{4}$ $8$ $4$ $12$ $( 1,14, 4,12)( 2,13, 3,11)( 5,10,16, 8)( 6, 9,15, 7)$
4J1 $4^{4}$ $8$ $4$ $12$ $( 1,10,15, 8)( 2, 9,16, 7)( 3,12, 5,14)( 4,11, 6,13)$
4J-1 $4^{4}$ $8$ $4$ $12$ $( 1,14,15,12)( 2,13,16,11)( 3, 8, 5,10)( 4, 7, 6, 9)$

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

32 x 32 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