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Properties

Label 20T16
20T16 1 9 1->9 16 1->16 1->16 2 10 2->10 15 2->15 2->15 3 3->10 3->15 17 3->17 4 4->9 4->16 18 4->18 5 5->2 5->3 20 5->20 6 6->1 6->4 19 6->19 7 7->1 8 7->8 7->17 8->2 8->18 9->4 12 9->12 14 9->14 10->3 11 10->11 13 10->13 11->5 11->5 11->19 12->6 12->6 12->20 13->5 13->8 13->19 14->6 14->7 14->20 15->9 15->11 15->13 16->10 16->12 16->14 17->11 17->18 18->12 19->2 19->3 19->13 20->1 20->4 20->14
Degree $20$
Order $80$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2^2\times F_5$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 16);
 
Copy content sage:G = TransitiveGroup(20, 16)
 
Copy content oscar:G = transitive_group(20, 16)
 
Copy content gap:G := TransitiveGroup(20, 16);
 

Group invariants

Abstract group:  $C_2^2\times F_5$
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:  $80=2^{4} \cdot 5$
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 $n$:  $20$
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 $t$:  $16$
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(20).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(20), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(20), G));
 
Generators:  $(1,16,14,20)(2,15,13,19)(3,10,11,5)(4,9,12,6)(7,17)(8,18)$, $(1,9,14,6)(2,10,13,5)(3,15,11,19)(4,16,12,20)(7,8)(17,18)$, $(1,16,10,3,17,11,5,20,14,7)(2,15,9,4,18,12,6,19,13,8)$
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
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_4\times C_2^2$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $F_5$

Degree 10: $F_{5}\times C_2$ x 3

Low degree siblings

20T16 x 3, 40T44 x 3, 40T56

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{20}$ $1$ $1$ $0$ $()$
2A $2^{10}$ $1$ $2$ $10$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,18)( 8,17)( 9,20)(10,19)$
2B $2^{10}$ $1$ $2$ $10$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,19)(10,20)$
2C $2^{10}$ $1$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2D $2^{10}$ $5$ $2$ $10$ $( 1,15)( 2,16)( 3,13)( 4,14)( 5,12)( 6,11)( 7, 9)( 8,10)(17,19)(18,20)$
2E $2^{10}$ $5$ $2$ $10$ $( 1, 3)( 2, 4)( 5,20)( 6,19)( 7,17)( 8,18)( 9,15)(10,16)(11,14)(12,13)$
2F $2^{10}$ $5$ $2$ $10$ $( 1, 6)( 2, 5)( 3, 4)( 7,19)( 8,20)( 9,17)(10,18)(11,15)(12,16)(13,14)$
2G $2^{8},1^{4}$ $5$ $2$ $8$ $( 1,14)( 2,13)( 3,11)( 4,12)( 5,10)( 6, 9)(15,19)(16,20)$
4A1 $4^{4},2^{2}$ $5$ $4$ $14$ $( 1,20,14,16)( 2,19,13,15)( 3, 5,11,10)( 4, 6,12, 9)( 7,17)( 8,18)$
4A-1 $4^{4},2^{2}$ $5$ $4$ $14$ $( 1,16,14,20)( 2,15,13,19)( 3,10,11, 5)( 4, 9,12, 6)( 7,17)( 8,18)$
4B1 $4^{4},2^{2}$ $5$ $4$ $14$ $( 1, 2)( 3, 8,20,15)( 4, 7,19,16)( 5,13,17, 9)( 6,14,18,10)(11,12)$
4B-1 $4^{4},2^{2}$ $5$ $4$ $14$ $( 1,18,10,13)( 2,17, 9,14)( 3,12, 7,19)( 4,11, 8,20)( 5, 6)(15,16)$
4C1 $4^{4},1^{4}$ $5$ $4$ $12$ $( 1,10,14, 5)( 2, 9,13, 6)( 3,16,11,20)( 4,15,12,19)$
4C-1 $4^{4},1^{4}$ $5$ $4$ $12$ $( 1, 5,14,10)( 2, 6,13, 9)( 3,20,11,16)( 4,19,12,15)$
4D1 $4^{4},2^{2}$ $5$ $4$ $14$ $( 1,12)( 2,11)( 3,18,20, 6)( 4,17,19, 5)( 7, 9,16,13)( 8,10,15,14)$
4D-1 $4^{4},2^{2}$ $5$ $4$ $14$ $( 1, 8,10, 4)( 2, 7, 9, 3)( 5,15)( 6,16)(11,18,20,13)(12,17,19,14)$
5A $5^{4}$ $4$ $5$ $16$ $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3, 7,11,16,20)( 4, 8,12,15,19)$
10A $10^{2}$ $4$ $10$ $18$ $( 1, 4, 5, 8,10,12,14,15,17,19)( 2, 3, 6, 7, 9,11,13,16,18,20)$
10B $10^{2}$ $4$ $10$ $18$ $( 1,16,10, 3,17,11, 5,20,14, 7)( 2,15, 9, 4,18,12, 6,19,13, 8)$
10C $10^{2}$ $4$ $10$ $18$ $( 1,13, 5,18,10, 2,14, 6,17, 9)( 3,15, 7,19,11, 4,16, 8,20,12)$

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 2C 2D 2E 2F 2G 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1 5A 10A 10B 10C
Size 1 1 1 1 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2G 2G 2G 2G 2G 2G 2G 2G 5A 5A 5A 5A
5 P 1A 2A 2B 2C 2D 2E 2F 2G 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1 1A 2A 2B 2C
Type
80.50.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.50.1b R 1 −1 −1 1 1 1 −1 −1 −1 1 −1 −1 1 1 −1 1 1 −1 1 −1
80.50.1c R 1 −1 −1 1 1 1 −1 −1 1 −1 1 1 −1 −1 1 −1 1 −1 1 −1
80.50.1d R 1 −1 1 −1 −1 1 −1 1 −1 1 1 −1 −1 −1 1 1 1 1 −1 −1
80.50.1e R 1 −1 1 −1 −1 1 −1 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1
80.50.1f R 1 1 −1 −1 −1 1 1 −1 −1 −1 1 −1 1 1 1 −1 1 −1 −1 1
80.50.1g R 1 1 −1 −1 −1 1 1 −1 1 1 −1 1 −1 −1 −1 1 1 −1 −1 1
80.50.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
80.50.1i1 C 1 −1 −1 1 −1 −1 1 1 −i −i −i i −i i i i 1 −1 1 −1
80.50.1i2 C 1 −1 −1 1 −1 −1 1 1 i i i −i i −i −i −i 1 −1 1 −1
80.50.1j1 C 1 −1 1 −1 1 −1 1 −1 −i −i i i i −i −i i 1 1 −1 −1
80.50.1j2 C 1 −1 1 −1 1 −1 1 −1 i i −i −i −i i i −i 1 1 −1 −1
80.50.1k1 C 1 1 −1 −1 1 −1 −1 1 −i i i i −i i −i −i 1 −1 −1 1
80.50.1k2 C 1 1 −1 −1 1 −1 −1 1 i −i −i −i i −i i i 1 −1 −1 1
80.50.1l1 C 1 1 1 1 −1 −1 −1 −1 −i i −i i i −i i −i 1 1 1 1
80.50.1l2 C 1 1 1 1 −1 −1 −1 −1 i −i i −i −i i −i i 1 1 1 1
80.50.4a R 4 4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1
80.50.4b R 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 −1 1
80.50.4c R 4 −4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1
80.50.4d R 4 4 −4 −4 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 1 −1

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