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Properties

Label 16T74
16T74 1 4 1->4 12 1->12 2 3 2->3 11 2->11 14 3->14 13 4->13 5 8 5->8 15 5->15 6 7 6->7 16 6->16 10 7->10 9 8->9 9->1 9->12 10->2 10->11 11->7 11->9 12->8 12->10 13->6 13->16 14->5 14->15 15->3 15->13 16->4 16->14
Degree $16$
Order $64$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_4^2:C_4$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(16, 74);
 
Copy content sage:G = TransitiveGroup(16, 74)
 
Copy content oscar:G = transitive_group(16, 74)
 
Copy content gap:G := TransitiveGroup(16, 74);
 

Group invariants

Abstract group:  $C_4^2:C_4$
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:  $64=2^{6}$
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 $t$:  $74$
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:  $(1,4)(2,3)(5,8)(6,7)(9,12,10,11)(13,16,14,15)$, $(1,12,8,9)(2,11,7,10)(3,14,5,15)(4,13,6,16)$
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 6, $C_2^2$
$8$:  $D_{4}$ x 3, $C_4\times C_2$ x 3, $Q_8$
$16$:  $C_2^2:C_4$ x 3, $C_4^2$, $C_4:C_4$ x 3
$32$:  32T41

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Degree 8: $C_2^2:C_4$

Low degree siblings

16T74 x 3, 16T123, 32T64 x 2, 32T131

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 $2^{8}$ $2$ $2$ $8$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,11)(10,12)(13,15)(14,16)$
2C $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2D $2^{8}$ $2$ $2$ $8$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,12)(10,11)(13,16)(14,15)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,14,10,13)(11,16,12,15)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,15,12,16)$
4C $4^{4}$ $2$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,10,16)(11,13,12,14)$
4D $4^{4}$ $2$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,16,10,15)(11,14,12,13)$
4E1 $4^{2},2^{4}$ $4$ $4$ $10$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12,10,11)(13,16,14,15)$
4E-1 $4^{2},2^{4}$ $4$ $4$ $10$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11,10,12)(13,15,14,16)$
4F1 $4^{4}$ $4$ $4$ $12$ $( 1,12, 8, 9)( 2,11, 7,10)( 3,14, 5,15)( 4,13, 6,16)$
4F-1 $4^{4}$ $4$ $4$ $12$ $( 1, 9, 8,12)( 2,10, 7,11)( 3,15, 5,14)( 4,16, 6,13)$
4G1 $4^{2},2^{2},1^{4}$ $4$ $4$ $8$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,10)(13,14)$
4G-1 $4^{2},2^{2},1^{4}$ $4$ $4$ $8$ $( 1, 5, 2, 6)( 3, 8, 4, 7)(11,12)(15,16)$
4H1 $4^{4}$ $4$ $4$ $12$ $( 1,11, 7, 9)( 2,12, 8,10)( 3,13, 6,15)( 4,14, 5,16)$
4H-1 $4^{4}$ $4$ $4$ $12$ $( 1,10, 7,12)( 2, 9, 8,11)( 3,16, 6,14)( 4,15, 5,13)$
8A1 $8^{2}$ $4$ $8$ $14$ $( 1,10, 3,15, 2, 9, 4,16)( 5,13, 7,11, 6,14, 8,12)$
8A-1 $8^{2}$ $4$ $8$ $14$ $( 1,11, 3,14, 2,12, 4,13)( 5,16, 7,10, 6,15, 8, 9)$
8B1 $8^{2}$ $4$ $8$ $14$ $( 1, 9, 4,15, 2,10, 3,16)( 5,14, 8,11, 6,13, 7,12)$
8B-1 $8^{2}$ $4$ $8$ $14$ $( 1,12, 4,14, 2,11, 3,13)( 5,15, 8,10, 6,16, 7, 9)$

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

1A 2A 2B 2C 2D 4A1 4A-1 4B 4C 4D 4E1 4E-1 4F1 4F-1 4G1 4G-1 4H1 4H-1 8A1 8A-1 8B1 8B-1
Size 1 1 2 2 2 1 1 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2C 2C 2D 2D 2C 2C 2D 2D 4D 4D 4D 4D
Type
64.18.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.18.1b R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
64.18.1c R 1 1 1 1 1 1 1 1 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1 −1 −1
64.18.1d R 1 1 1 1 1 1 1 1 1 1 1 −1 1 1 −1 −1 −1 1 −1 −1 −1 −1
64.18.1e1 C 1 1 −1 −1 1 1 1 −1 −1 1 −i −i i −i i i −i i −1 1 −1 1
64.18.1e2 C 1 1 −1 −1 1 1 1 −1 −1 1 i i −i i −i −i i −i −1 1 −1 1
64.18.1f1 C 1 1 −1 −1 1 1 1 −1 −1 1 −i i i −i −i −i i i 1 −1 1 −1
64.18.1f2 C 1 1 −1 −1 1 1 1 −1 −1 1 i −i −i i i i −i −i 1 −1 1 −1
64.18.1g1 C 1 1 −1 1 −1 1 1 −1 1 −1 −1 −i −1 1 −i i i 1 i i −i −i
64.18.1g2 C 1 1 −1 1 −1 1 1 −1 1 −1 −1 i −1 1 i −i −i 1 −i −i i i
64.18.1h1 C 1 1 −1 1 −1 1 1 −1 1 −1 1 −i 1 −1 −i i i −1 −i −i i i
64.18.1h2 C 1 1 −1 1 −1 1 1 −1 1 −1 1 i 1 −1 i −i −i −1 i i −i −i
64.18.1i1 C 1 1 1 −1 −1 1 1 1 −1 −1 −i −1 i i 1 −1 1 −i i −i −i i
64.18.1i2 C 1 1 1 −1 −1 1 1 1 −1 −1 i −1 −i −i 1 −1 1 i −i i i −i
64.18.1j1 C 1 1 1 −1 −1 1 1 1 −1 −1 −i 1 i i −1 1 −1 −i −i i i −i
64.18.1j2 C 1 1 1 −1 −1 1 1 1 −1 −1 i 1 −i −i −1 1 −1 i i −i −i i
64.18.2a R 2 2 −2 2 −2 −2 −2 2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
64.18.2b R 2 2 2 −2 −2 −2 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
64.18.2c R 2 2 2 2 2 −2 −2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
64.18.2d S 2 2 −2 −2 2 −2 −2 2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
64.18.4a1 C 4 −4 0 0 0 −4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
64.18.4a2 C 4 −4 0 0 0 4i −4i 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