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Properties

Label 16T1275
16T1275 1 2 1->2 5 1->5 8 1->8 6 2->6 7 2->7 3 4 3->4 3->5 3->8 4->6 4->7 5->2 11 5->11 16 5->16 6->1 12 6->12 15 6->15 7->3 9 7->9 14 7->14 8->4 10 8->10 13 8->13 9->1 9->8 9->16 10->2 10->7 10->15 11->3 11->6 11->13 12->4 12->5 12->14 13->10 13->11 13->15 14->9 14->12 14->16 15->10 15->11 16->9 16->12
Degree $16$
Order $1024$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2^6:D_8$

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, 1275);
 
Copy content sage:G = TransitiveGroup(16, 1275)
 
Copy content oscar:G = transitive_group(16, 1275)
 
Copy content gap:G := TransitiveGroup(16, 1275);
 

Group invariants

Abstract group:  $C_2^6:D_8$
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:  $1024=2^{10}$
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:  $6$
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 $t$:  $1275$
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:  $(1,2)(3,4)(5,11,6,12)(7,9,8,10)(13,15)(14,16)$, $(1,8,4,6)(2,7,3,5)(9,16,12,14)(10,15,11,13)$, $(1,5,16,9)(2,6,15,10)(3,8,13,11)(4,7,14,12)$
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
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_{8}$ x 2, $D_4\times C_2$ x 3
$32$:  $Z_8 : Z_8^\times$, $C_2^2 \wr C_2$, 16T29
$64$:  $(C_4^2 : C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T126
$128$:  $C_2 \wr C_2\wr C_2$ x 2, 16T409
$256$:  16T689
$512$:  16T962

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2 \wr C_2\wr C_2$

Low degree siblings

16T1253 x 8, 16T1265 x 8, 16T1275 x 15, 32T36646 x 4, 32T36647 x 8, 32T36648 x 16, 32T36649 x 4, 32T36650 x 8, 32T36651 x 8, 32T36652 x 8, 32T36653 x 8, 32T36733 x 8, 32T36734 x 8, 32T36735 x 8, 32T36736 x 8, 32T36737 x 8, 32T36738 x 8, 32T36739 x 8, 32T36740 x 4, 32T36741 x 8, 32T36742 x 8, 32T36743 x 4, 32T36744 x 4, 32T36745 x 16, 32T36746 x 4, 32T36747 x 4, 32T36748 x 8, 32T36749 x 4, 32T36750 x 8, 32T36820 x 8, 32T36821 x 8, 32T36822 x 8, 32T36823 x 8, 32T36824 x 8, 32T36825 x 8, 32T41957 x 8, 32T56530 x 4

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 2)( 3, 4)(13,14)(15,16)$
2C $2^{4},1^{8}$ $4$ $2$ $4$ $( 5, 8)( 6, 7)( 9,11)(10,12)$
2D $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,14)(15,16)$
2E $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$
2F $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2G $2^{4},1^{8}$ $4$ $2$ $4$ $( 5, 6)( 7, 8)(13,14)(15,16)$
2H $2^{8}$ $8$ $2$ $8$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5, 6)( 7, 8)( 9,10)(11,12)$
2I $2^{8}$ $8$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 9)( 6,10)( 7,12)( 8,11)(13,15)(14,16)$
2J $2^{8}$ $8$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,16)(14,15)$
2K $2^{4},1^{8}$ $8$ $2$ $4$ $( 5, 9)( 6,10)( 7,12)( 8,11)$
2L $2^{8}$ $8$ $2$ $8$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5, 9)( 6,10)( 7,12)( 8,11)$
2M $2^{8}$ $8$ $2$ $8$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5, 9)( 6,10)( 7,12)( 8,11)$
2N $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)(13,15)(14,16)$
2O $2^{6},1^{4}$ $16$ $2$ $6$ $( 1,16)( 2,15)( 3,14)( 4,13)( 9,11)(10,12)$
2P $2^{6},1^{4}$ $16$ $2$ $6$ $( 1, 2)( 5, 6)( 7, 8)( 9,12)(10,11)(15,16)$
2Q $2^{8}$ $16$ $2$ $8$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5, 7)( 6, 8)( 9,10)(11,12)$
2R $2^{4},1^{8}$ $16$ $2$ $4$ $( 1, 2)( 5, 8)( 6, 7)(15,16)$
2S $2^{8}$ $32$ $2$ $8$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,15)( 6,16)( 7,13)( 8,14)$
4A $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 3, 2, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,15,14,16)$
4B $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1,14, 2,13)( 3,16, 4,15)( 9,11)(10,12)$
4C $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1, 3, 2, 4)( 5, 8)( 6, 7)(13,15,14,16)$
4D $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,14, 2,13)( 3,16, 4,15)( 5, 7)( 6, 8)( 9,10)(11,12)$
4E $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1,15, 2,16)( 3,14, 4,13)( 5, 6)( 7, 8)$
4F $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 3)( 2, 4)( 5,10, 6, 9)( 7,11, 8,12)(13,15)(14,16)$
4G $4^{4}$ $16$ $4$ $12$ $( 1,16, 2,15)( 3,13, 4,14)( 5, 9, 6,10)( 7,12, 8,11)$
4H $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,15)( 2,16)( 3,13)( 4,14)( 5,11, 8, 9)( 6,12, 7,10)$
4I $4^{4}$ $32$ $4$ $12$ $( 1, 4, 2, 3)( 5,12, 8,10)( 6,11, 7, 9)(13,16,14,15)$
4J $4^{4}$ $32$ $4$ $12$ $( 1,13, 2,14)( 3,15, 4,16)( 5,11, 8, 9)( 6,12, 7,10)$
4K $4^{2},2^{2},1^{4}$ $32$ $4$ $8$ $( 3, 4)( 5,12, 8,10)( 6,11, 7, 9)(13,14)$
4L $4^{4}$ $32$ $4$ $12$ $( 1,11, 3, 9)( 2,12, 4,10)( 5,15, 8,14)( 6,16, 7,13)$
4M $4^{4}$ $32$ $4$ $12$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,15, 6,16)( 7,13, 8,14)$
4N $4^{4}$ $32$ $4$ $12$ $( 1,12, 4, 9)( 2,11, 3,10)( 5,15, 7,13)( 6,16, 8,14)$
4O $4^{4}$ $32$ $4$ $12$ $( 1,14, 3,16)( 2,13, 4,15)( 5,12, 7,10)( 6,11, 8, 9)$
4P $4^{2},2^{2},1^{4}$ $32$ $4$ $8$ $( 3, 4)( 5, 7, 6, 8)(11,12)(13,15,14,16)$
4Q $4^{4}$ $64$ $4$ $12$ $( 1, 6,13,10)( 2, 5,14, 9)( 3, 8,16,11)( 4, 7,15,12)$
4R $4^{4}$ $64$ $4$ $12$ $( 1,11,15, 8)( 2,12,16, 7)( 3, 9,14, 5)( 4,10,13, 6)$
4S $4^{3},2,1^{2}$ $64$ $4$ $10$ $( 1,16, 3,14)( 2,15, 4,13)( 5, 8, 6, 7)( 9,10)$
8A1 $8^{2}$ $64$ $8$ $14$ $( 1,10,14, 5, 3,12,16, 7)( 2, 9,13, 6, 4,11,15, 8)$
8A3 $8^{2}$ $64$ $8$ $14$ $( 1, 5,16,10, 3, 7,14,12)( 2, 6,15, 9, 4, 8,13,11)$
8B1 $8,4,2^{2}$ $64$ $8$ $12$ $( 1, 9)( 2,10)( 3,12, 4,11)( 5,13, 7,15, 6,14, 8,16)$
8B3 $8,4,2^{2}$ $64$ $8$ $12$ $( 1, 9)( 2,10)( 3,11, 4,12)( 5,15, 8,13, 6,16, 7,14)$

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

43 x 43 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