↑

Properties

Label 20T48
20T48 1 2 1->2 7 1->7 8 2->8 3 6 3->6 19 3->19 4 5 4->5 20 4->20 5->3 17 5->17 6->4 18 6->18 7->4 9 7->9 8->3 10 8->10 14 9->14 9->19 13 10->13 10->20 11 11->7 11->13 12 12->8 12->14 16 13->16 15 14->15 15->11 15->18 16->12 16->17 17->11 18->12 19->1 19->16 20->2 20->15
Degree $20$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5\wr C_2$

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 48);
 
Copy content sage:G = TransitiveGroup(20, 48)
 
Copy content oscar:G = transitive_group(20, 48)
 

Group invariants

Abstract group:  $D_5\wr C_2$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $200=2^{3} \cdot 5^{2}$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar: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)
 
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)
 
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)
 
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
 

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)
 
Transitive number $t$:  $48$
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]
 
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)
 
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)
 
$\card{\Aut(F/K)}$:  $10$
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])
 
Generators:  $(1,7,9,19)(2,8,10,20)(3,6,4,5)(11,13,16,17)(12,14,15,18)$, $(1,2)(3,19,16,12,8)(4,20,15,11,7)(5,17)(6,18)(9,14)(10,13)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: None

Degree 10: $D_5^2 : C_2$

Low degree siblings

10T19, 10T21 x 2, 20T48, 20T50 x 2, 20T55, 20T57 x 2, 25T21, 40T167 x 2, 40T170

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 $2^{5},1^{10}$ $10$ $2$ $5$ $( 1, 2)( 5,17)( 6,18)( 9,14)(10,13)$
2B $2^{10}$ $10$ $2$ $10$ $( 1,15)( 2,16)( 3,13)( 4,14)( 5,12)( 6,11)( 7,10)( 8, 9)(17,20)(18,19)$
2C $2^{10}$ $25$ $2$ $10$ $( 1,18)( 2,17)( 3,15)( 4,16)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,20)$
4A $4^{5}$ $50$ $4$ $15$ $( 1, 7,18,12)( 2, 8,17,11)( 3,14,15, 5)( 4,13,16, 6)( 9,19,10,20)$
5A1 $5^{2},1^{10}$ $4$ $5$ $8$ $( 3, 8,12,16,19)( 4, 7,11,15,20)$
5A2 $5^{2},1^{10}$ $4$ $5$ $8$ $( 3,12,19, 8,16)( 4,11,20, 7,15)$
5B1 $5^{4}$ $4$ $5$ $16$ $( 1,14, 6,17,10)( 2,13, 5,18, 9)( 3,12,19, 8,16)( 4,11,20, 7,15)$
5B2 $5^{4}$ $4$ $5$ $16$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,19,16,12, 8)( 4,20,15,11, 7)$
5C $5^{4}$ $8$ $5$ $16$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,16, 8,19,12)( 4,15, 7,20,11)$
10A1 $5^{2},2^{5}$ $20$ $10$ $13$ $( 1, 2)( 3,16, 8,19,12)( 4,15, 7,20,11)( 5,17)( 6,18)( 9,14)(10,13)$
10A3 $5^{2},2^{5}$ $20$ $10$ $13$ $( 1, 2)( 3,19,16,12, 8)( 4,20,15,11, 7)( 5,17)( 6,18)( 9,14)(10,13)$
10B1 $10^{2}$ $20$ $10$ $18$ $( 1,20,14, 7, 6,15,17, 4,10,11)( 2,19,13, 8, 5,16,18, 3, 9,12)$
10B3 $10^{2}$ $20$ $10$ $18$ $( 1, 7,17,11,14,15,10,20, 6, 4)( 2, 8,18,12,13,16, 9,19, 5, 3)$

Malle's constant $a(G)$:     $1/5$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 

Character table

1A 2A 2B 2C 4A 5A1 5A2 5B1 5B2 5C 10A1 10A3 10B1 10B3
Size 1 10 10 25 50 4 4 4 4 8 20 20 20 20
2 P 1A 1A 1A 1A 2C 5A2 5A1 5B2 5B1 5C 5A1 5A2 5B1 5B2
5 P 1A 2A 2B 2C 4A 1A 1A 1A 1A 1A 2A 2A 2B 2B
Type
200.43.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
200.43.1b R 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
200.43.1c R 1 −1 1 1 −1 1 1 1 1 1 1 −1 1 −1
200.43.1d R 1 1 −1 1 −1 1 1 1 1 1 −1 1 −1 1
200.43.2a R 2 0 0 −2 0 2 2 2 2 2 0 0 0 0
200.43.4a1 R 4 0 2 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 ζ5−2+ζ52 0 ζ5−1+ζ5 0
200.43.4a2 R 4 0 2 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 ζ5−1+ζ5 0 ζ5−2+ζ52 0
200.43.4b1 R 4 2 0 0 0 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 0 ζ5−1+ζ5 0 ζ5−2+ζ52
200.43.4b2 R 4 2 0 0 0 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 0 ζ5−2+ζ52 0 ζ5−1+ζ5
200.43.4c1 R 4 −2 0 0 0 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 0 −ζ5−1−ζ5 0 −ζ5−2−ζ52
200.43.4c2 R 4 −2 0 0 0 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 0 −ζ5−2−ζ52 0 −ζ5−1−ζ5
200.43.4d1 R 4 0 −2 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 −ζ5−2−ζ52 0 −ζ5−1−ζ5 0
200.43.4d2 R 4 0 −2 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 −ζ5−1−ζ5 0 −ζ5−2−ζ52 0
200.43.8a R 8 0 0 0 0 −2 −2 −2 −2 3 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)
 

Regular extensions

Data not computed