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Properties

Label 16T1476
16T1476 1 11 1->11 16 1->16 2 12 2->12 15 2->15 3 9 3->9 14 3->14 4 10 4->10 13 4->13 5 7 5->7 5->16 6 8 6->8 6->15 7->13 8->14 9->10 13->2 13->6 14->1 14->5 15->4 15->7 16->3 16->8
Degree $16$
Order $2048$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2^7.D_8$

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, 1476);
 
Copy content sage:G = TransitiveGroup(16, 1476)
 
Copy content oscar:G = transitive_group(16, 1476)
 
Copy content gap:G := TransitiveGroup(16, 1476);
 

Group invariants

Abstract group:  $C_2^7.D_8$
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:  $2048=2^{11}$
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 $t$:  $1476$
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:  $(1,11)(2,12)(3,9)(4,10)(5,16,8,14)(6,15,7,13)$, $(1,16,3,14)(2,15,4,13)(5,7)(6,8)(9,10)$
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, $C_4\times C_2$
$16$:  $D_{8}$, $QD_{16}$, $C_2^2:C_4$
$32$:  $C_4\wr C_2$, $C_2^3 : C_4 $, 16T26
$64$:  $((C_8 : C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T163
$128$:  16T330
$256$:  16T682
$512$:  16T944
$1024$:  16T1272

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

16T1465 x 8, 16T1476 x 7, 32T99256 x 4, 32T99257 x 16, 32T99258 x 16, 32T99259 x 4, 32T99260 x 4, 32T99261 x 8, 32T99262 x 4, 32T99263 x 4, 32T99264 x 4, 32T99342 x 16, 32T99343 x 4, 32T99344 x 4, 32T99345 x 4, 32T99346 x 16, 32T99347 x 4, 32T99348 x 4, 32T99349 x 4, 32T104646 x 8, 32T104649 x 8, 32T116504 x 4, 32T116517 x 4, 32T117461 x 4, 32T117464 x 4, 32T144825 x 2, 32T145090 x 2, 32T182661 x 2, 32T182687 x 2, 32T192445 x 2, 32T204559 x 2

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
2C $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
2D $2^{2},1^{12}$ $4$ $2$ $2$ $(5,6)(7,8)$
2E $2^{4},1^{8}$ $4$ $2$ $4$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
2F $2^{8}$ $8$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
2G $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$
2H $2^{8}$ $8$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
2I $2^{8}$ $8$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)$
2J $2^{8}$ $8$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 8)( 6, 7)( 9,10)(11,12)(13,16)(14,15)$
2K $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 3)( 2, 4)( 9,10)(11,12)(13,15)(14,16)$
2L $2^{4},1^{8}$ $8$ $2$ $4$ $( 5, 7)( 6, 8)( 9,12)(10,11)$
2M $2^{4},1^{8}$ $8$ $2$ $4$ $( 9,12)(10,11)(13,15)(14,16)$
2N $2^{8}$ $8$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,16)(14,15)$
2O $2^{4},1^{8}$ $8$ $2$ $4$ $(1,3)(2,4)(5,8)(6,7)$
2P $2^{6},1^{4}$ $16$ $2$ $6$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,11)(10,12)$
2Q $2^{6},1^{4}$ $16$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)(13,16)(14,15)$
2R $2^{4},1^{8}$ $16$ $2$ $4$ $( 1, 2)( 5, 6)( 9,10)(13,14)$
2S $2^{5},1^{6}$ $32$ $2$ $5$ $( 3, 4)( 5,11)( 6,12)( 7,10)( 8, 9)$
2T $2^{7},1^{2}$ $32$ $2$ $7$ $( 3, 4)( 5,11)( 6,12)( 7,10)( 8, 9)(13,14)(15,16)$
4A $4^{4}$ $16$ $4$ $12$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$
4B $4^{2},2,1^{6}$ $32$ $4$ $7$ $( 3, 4)( 5,10, 6, 9)( 7,11, 8,12)$
4C $4^{2},2^{3},1^{2}$ $32$ $4$ $9$ $( 3, 4)( 5,10, 6, 9)( 7,11, 8,12)(13,14)(15,16)$
4D $4^{2},2^{2},1^{4}$ $32$ $4$ $8$ $( 3, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,14)$
4E $4^{3},1^{4}$ $64$ $4$ $9$ $( 1,14, 3,16)( 2,13, 4,15)( 9,12,10,11)$
4F $4^{2},2^{3},1^{2}$ $64$ $4$ $9$ $( 1, 2)( 5,10, 7,11)( 6, 9, 8,12)(13,15)(14,16)$
4G $4^{3},2^{2}$ $64$ $4$ $11$ $( 1, 4, 2, 3)( 5, 9, 6,10)( 7,12, 8,11)(13,16)(14,15)$
4H $4^{3},2^{2}$ $64$ $4$ $11$ $( 1,14, 4,16)( 2,13, 3,15)( 5, 7, 6, 8)( 9,10)(11,12)$
4I $4,2^{6}$ $64$ $4$ $9$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5, 7, 6, 8)( 9,11)(10,12)$
4J $4^{2},2^{3},1^{2}$ $64$ $4$ $9$ $( 1, 3)( 2, 4)( 5, 9, 8,12)( 6,10, 7,11)(15,16)$
4K $4^{2},2^{2},1^{4}$ $64$ $4$ $8$ $( 1, 4, 2, 3)( 7, 8)( 9,12,10,11)(13,14)$
4L1 $4^{4}$ $64$ $4$ $12$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,16, 8,13)( 6,15, 7,14)$
4L-1 $4^{4}$ $64$ $4$ $12$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 8,16)( 6,14, 7,15)$
4M1 $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,14,12,16)(10,13,11,15)$
4M-1 $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,16,12,14)(10,15,11,13)$
4N1 $4^{4}$ $64$ $4$ $12$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,13, 7,16)( 6,14, 8,15)$
4N-1 $4^{4}$ $64$ $4$ $12$ $( 1,12, 2,11)( 3,10, 4, 9)( 5,16, 7,13)( 6,15, 8,14)$
4O1 $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,14)(10,13)(11,15)(12,16)$
4O-1 $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 8, 3, 5)( 2, 7, 4, 6)( 9,14)(10,13)(11,15)(12,16)$
4P $4^{2},2^{4}$ $128$ $4$ $10$ $( 1,14, 2,13)( 3,16)( 4,15)( 5,10, 6, 9)( 7,11)( 8,12)$
8A1 $8^{2}$ $64$ $8$ $14$ $( 1,13, 4,16, 2,14, 3,15)( 5,12, 7, 9, 6,11, 8,10)$
8A-1 $8^{2}$ $64$ $8$ $14$ $( 1,15, 3,14, 2,16, 4,13)( 5,10, 8,11, 6, 9, 7,12)$
16A1 $16$ $128$ $16$ $15$ $( 1, 7,13, 9, 4, 6,16,11, 2, 8,14,10, 3, 5,15,12)$
16A-1 $16$ $128$ $16$ $15$ $( 1,12,15, 5, 3,10,14, 8, 2,11,16, 6, 4, 9,13, 7)$
16A3 $16$ $128$ $16$ $15$ $( 1, 9,16, 8, 3,12,13, 6, 2,10,15, 7, 4,11,14, 5)$
16A-3 $16$ $128$ $16$ $15$ $( 1, 5,14,11, 4, 7,15,10, 2, 6,13,12, 3, 8,16, 9)$

Malle's constant $a(G)$:     $1/2$

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

47 x 47 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