Properties

Label 16T60
16T60 1 13 1->13 14 1->14 2 2->13 2->14 3 8 3->8 10 3->10 4 7 4->7 9 4->9 5 5->10 11 5->11 6 6->9 12 6->12 7->1 8->2 9->3 10->4 15 11->15 16 11->16 12->15 12->16 13->6 14->5 15->7 16->8
Degree $16$
Order $48$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $\SL(2,3):C_2$

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, 60);
 
Copy content sage:G = TransitiveGroup(16, 60)
 
Copy content oscar:G = transitive_group(16, 60)
 
Copy content gap:G := TransitiveGroup(16, 60);
 

Group invariants

Abstract group:  $\SL(2,3):C_2$
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:  $48=2^{4} \cdot 3$
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:   not nilpotent
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 $t$:  $60$
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:  $(1,14,5,11,16,8,2,13,6,12,15,7)(3,10,4,9)$, $(1,13)(2,14)(3,8)(4,7)(5,10)(6,9)(11,15)(12,16)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $A_4$

Degree 8: $A_4\times C_2$

Low degree siblings

24T21

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 $2^{8}$ $6$ $2$ $8$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,15)( 8,16)$
3A1 $3^{4},1^{4}$ $4$ $3$ $8$ $( 3, 6,15)( 4, 5,16)( 7,14, 9)( 8,13,10)$
3A-1 $3^{4},1^{4}$ $4$ $3$ $8$ $( 3,15, 6)( 4,16, 5)( 7, 9,14)( 8,10,13)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1,11, 2,12)( 3, 9, 4,10)( 5, 8, 6, 7)(13,15,14,16)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1,12, 2,11)( 3,10, 4, 9)( 5, 7, 6, 8)(13,16,14,15)$
4B $4^{4}$ $6$ $4$ $12$ $( 1, 6, 2, 5)( 3,16, 4,15)( 7,12, 8,11)( 9,13,10,14)$
6A1 $6^{2},2^{2}$ $4$ $6$ $12$ $( 1, 2)( 3,16, 6, 4,15, 5)( 7,10,14, 8, 9,13)(11,12)$
6A-1 $6^{2},2^{2}$ $4$ $6$ $12$ $( 1, 2)( 3, 5,15, 4, 6,16)( 7,13, 9, 8,14,10)(11,12)$
12A1 $12,4$ $4$ $12$ $14$ $( 1,12, 2,11)( 3, 8,16, 9, 6,13, 4, 7,15,10, 5,14)$
12A-1 $12,4$ $4$ $12$ $14$ $( 1,11, 2,12)( 3,14, 5,10,15, 7, 4,13, 6, 9,16, 8)$
12A5 $12,4$ $4$ $12$ $14$ $( 1, 8,15,11, 6,14, 2, 7,16,12, 5,13)( 3,10, 4, 9)$
12A-5 $12,4$ $4$ $12$ $14$ $( 1,13, 5,12,16, 7, 2,14, 6,11,15, 8)( 3, 9, 4,10)$

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 2B 3A1 3A-1 4A1 4A-1 4B 6A1 6A-1 12A1 12A-1 12A5 12A-5
Size 1 1 6 4 4 1 1 6 4 4 4 4 4 4
2 P 1A 1A 1A 3A-1 3A1 2A 2A 2A 3A1 3A-1 6A1 6A-1 6A-1 6A1
3 P 1A 2A 2B 1A 1A 4A-1 4A1 4B 2A 2A 4A1 4A-1 4A1 4A-1
Type
48.33.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.33.1b R 1 1 −1 1 1 −1 −1 1 1 1 −1 −1 −1 −1
48.33.1c1 C 1 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
48.33.1c2 C 1 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
48.33.1d1 C 1 1 −1 ζ3−1 ζ3 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
48.33.1d2 C 1 1 −1 ζ3 ζ3−1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
48.33.2a1 C 2 −2 0 −1 −1 −2i 2i 0 1 1 −i i −i i
48.33.2a2 C 2 −2 0 −1 −1 2i −2i 0 1 1 i −i i −i
48.33.2b1 C 2 −2 0 −ζ124 ζ122 −2ζ123 2ζ123 0 −ζ122 ζ124 ζ12 −ζ125 ζ125 −ζ12
48.33.2b2 C 2 −2 0 ζ122 −ζ124 2ζ123 −2ζ123 0 ζ124 −ζ122 −ζ125 ζ12 −ζ12 ζ125
48.33.2b3 C 2 −2 0 −ζ124 ζ122 2ζ123 −2ζ123 0 −ζ122 ζ124 −ζ12 ζ125 −ζ125 ζ12
48.33.2b4 C 2 −2 0 ζ122 −ζ124 −2ζ123 2ζ123 0 ζ124 −ζ122 ζ125 −ζ12 ζ12 −ζ125
48.33.3a R 3 3 −1 0 0 3 3 −1 0 0 0 0 0 0
48.33.3b R 3 3 1 0 0 −3 −3 −1 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