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Properties

Label 16T1
16T1 1 7 1->7 2 8 2->8 3 10 3->10 4 9 4->9 5 11 5->11 6 12 6->12 14 7->14 13 8->13 15 9->15 16 10->16 11->2 12->1 13->3 14->4 15->5 16->6
Degree $16$
Order $16$
Cyclic yes
Abelian yes
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_{16}$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(16, 1);
 
Copy content sage:G = TransitiveGroup(16, 1)
 
Copy content oscar:G = transitive_group(16, 1)
 
Copy content gap:G := TransitiveGroup(16, 1);
 

Group invariants

Abstract group:  $C_{16}$
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:  $16=2^{4}$
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:  yes
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:  yes
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:  $1$
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 $t$:  $1$
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);
 
$\card{\Aut(F/K)}$:  $16$
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:  $(1,7,14,4,9,15,5,11,2,8,13,3,10,16,6,12)$
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)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

Low degree siblings

There are no siblings 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 $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,15, 8,16)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,14, 6,13)( 7,16, 8,15)$
8A1 $8^{2}$ $1$ $8$ $14$ $( 1,14, 9, 5, 2,13,10, 6)( 3,16,12, 7, 4,15,11, 8)$
8A-1 $8^{2}$ $1$ $8$ $14$ $( 1, 6,10,13, 2, 5, 9,14)( 3, 8,11,15, 4, 7,12,16)$
8A3 $8^{2}$ $1$ $8$ $14$ $( 1, 5,10,14, 2, 6, 9,13)( 3, 7,11,16, 4, 8,12,15)$
8A-3 $8^{2}$ $1$ $8$ $14$ $( 1,13, 9, 6, 2,14,10, 5)( 3,15,12, 8, 4,16,11, 7)$
16A1 $16$ $1$ $16$ $15$ $( 1, 7,14, 4, 9,15, 5,11, 2, 8,13, 3,10,16, 6,12)$
16A-1 $16$ $1$ $16$ $15$ $( 1,12, 6,16,10, 3,13, 8, 2,11, 5,15, 9, 4,14, 7)$
16A3 $16$ $1$ $16$ $15$ $( 1, 4, 5, 8,10,12,14,15, 2, 3, 6, 7, 9,11,13,16)$
16A-3 $16$ $1$ $16$ $15$ $( 1,16,13,11, 9, 7, 6, 3, 2,15,14,12,10, 8, 5, 4)$
16A5 $16$ $1$ $16$ $15$ $( 1,15,13,12, 9, 8, 6, 4, 2,16,14,11,10, 7, 5, 3)$
16A-5 $16$ $1$ $16$ $15$ $( 1, 3, 5, 7,10,11,14,16, 2, 4, 6, 8, 9,12,13,15)$
16A7 $16$ $1$ $16$ $15$ $( 1,11, 6,15,10, 4,13, 7, 2,12, 5,16, 9, 3,14, 8)$
16A-7 $16$ $1$ $16$ $15$ $( 1, 8,14, 3, 9,16, 5,12, 2, 7,13, 4,10,15, 6,11)$

Malle's constant $a(G)$:     $1/8$

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 4A1 4A-1 8A1 8A-1 8A3 8A-3 16A1 16A-1 16A3 16A-3 16A5 16A-5 16A7 16A-7
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 2A 2A 4A1 4A-1 4A-1 4A1 8A1 8A-1 8A3 8A-3 8A-3 8A3 8A-1 8A1
Type
16.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16.1.1b R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
16.1.1c1 C 1 1 1 1 −1 −1 −1 −1 −i i i −i −i i i −i
16.1.1c2 C 1 1 1 1 −1 −1 −1 −1 i −i −i i i −i −i i
16.1.1d1 C 1 1 −1 −1 −ζ82 ζ82 ζ82 −ζ82 ζ83 −ζ8 ζ8 −ζ83 −ζ83 ζ8 −ζ8 ζ83
16.1.1d2 C 1 1 −1 −1 ζ82 −ζ82 −ζ82 ζ82 −ζ8 ζ83 −ζ83 ζ8 ζ8 −ζ83 ζ83 −ζ8
16.1.1d3 C 1 1 −1 −1 −ζ82 ζ82 ζ82 −ζ82 −ζ83 ζ8 −ζ8 ζ83 ζ83 −ζ8 ζ8 −ζ83
16.1.1d4 C 1 1 −1 −1 ζ82 −ζ82 −ζ82 ζ82 ζ8 −ζ83 ζ83 −ζ8 −ζ8 ζ83 −ζ83 ζ8
16.1.1e1 C 1 −1 −ζ164 ζ164 ζ166 −ζ162 ζ162 −ζ166 −ζ163 ζ165 ζ16 −ζ167 ζ167 −ζ16 −ζ165 ζ163
16.1.1e2 C 1 −1 ζ164 −ζ164 −ζ162 ζ166 −ζ166 ζ162 ζ165 −ζ163 −ζ167 ζ16 −ζ16 ζ167 ζ163 −ζ165
16.1.1e3 C 1 −1 −ζ164 ζ164 ζ166 −ζ162 ζ162 −ζ166 ζ163 −ζ165 −ζ16 ζ167 −ζ167 ζ16 ζ165 −ζ163
16.1.1e4 C 1 −1 ζ164 −ζ164 −ζ162 ζ166 −ζ166 ζ162 −ζ165 ζ163 ζ167 −ζ16 ζ16 −ζ167 −ζ163 ζ165
16.1.1e5 C 1 −1 −ζ164 ζ164 −ζ166 ζ162 −ζ162 ζ166 ζ167 −ζ16 ζ165 −ζ163 ζ163 −ζ165 ζ16 −ζ167
16.1.1e6 C 1 −1 ζ164 −ζ164 ζ162 −ζ166 ζ166 −ζ162 −ζ16 ζ167 −ζ163 ζ165 −ζ165 ζ163 −ζ167 ζ16
16.1.1e7 C 1 −1 −ζ164 ζ164 −ζ166 ζ162 −ζ162 ζ166 −ζ167 ζ16 −ζ165 ζ163 −ζ163 ζ165 −ζ16 ζ167
16.1.1e8 C 1 −1 ζ164 −ζ164 ζ162 −ζ166 ζ166 −ζ162 ζ16 −ζ167 ζ163 −ζ165 ζ165 −ζ163 ζ167 −ζ16

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

$f_{ 1 } =$ $x^{16} + \left(-16 t^{8} - 16\right) x^{14} + \left(256 t^{14} - 224 t^{12} + 96 t^{10} + 80 t^{8} + 256 t^{6} - 224 t^{4} + 96 t^{2} + 80\right) x^{12} + \left(-1536 t^{20} + 1792 t^{18} - 384 t^{16} - 2304 t^{14} + 448 t^{12} + 960 t^{10} - 512 t^{8} - 2304 t^{6} + 1984 t^{4} - 832 t^{2} - 128\right) x^{10} + \left(4096 t^{26} - 3840 t^{24} - 4608 t^{22} + 17920 t^{20} - 11520 t^{18} + 768 t^{16} + 640 t^{14} + 13568 t^{12} - 13824 t^{10} + 4640 t^{8} + 5248 t^{6} - 4352 t^{4} + 1792 t^{2} + 32\right) x^{8} + \left(-4096 t^{32} - 4096 t^{30} + 30720 t^{28} - 53248 t^{26} + 23552 t^{24} + 12288 t^{22} - 11264 t^{20} - 20480 t^{18} + 8960 t^{16} + 21504 t^{14} - 42752 t^{12} + 32256 t^{10} - 18688 t^{8} + 5120 t^{6} - 768 t^{4} - 512 t^{2}\right) x^{6} + \left(16384 t^{36} - 49152 t^{34} + 53248 t^{32} + 20480 t^{30} - 94208 t^{28} + 83968 t^{26} - 27648 t^{24} + 49152 t^{22} - 105984 t^{20} + 116224 t^{18} - 63744 t^{16} + 22016 t^{14} + 5888 t^{12} - 16896 t^{10} + 17152 t^{8} - 6656 t^{6} + 1280 t^{4}\right) x^{4} + \left(-32768 t^{36} + 114688 t^{34} - 163840 t^{32} + 73728 t^{30} + 53248 t^{28} - 32768 t^{26} - 75776 t^{24} + 45056 t^{22} + 103424 t^{20} - 164864 t^{18} + 95232 t^{16} - 29696 t^{14} + 17408 t^{12} - 17408 t^{10} + 7168 t^{8} - 1024 t^{6}\right) x^{2} + \left(16384 t^{36} - 65536 t^{34} + 114688 t^{32} - 98304 t^{30} + 36864 t^{28} - 28672 t^{26} + 81920 t^{24} - 92160 t^{22} + 24576 t^{20} + 34816 t^{18} - 32512 t^{16} + 6144 t^{14} + 4096 t^{12} - 2048 t^{10} + 256 t^{8}\right)$ Copy content Toggle raw display