Properties

Label 20T125
20T125 1 4 1->4 17 1->17 2 5 2->5 16 2->16 3 3->1 20 3->20 4->2 19 4->19 5->3 18 5->18 6 6->1 7 6->7 7->2 8 7->8 8->3 9 8->9 9->4 10 9->10 10->5 10->6 11 11->10 12 12->6 13 13->7 14 14->8 15 15->9 16->13 16->20 17->12 17->16 18->11 18->17 19->15 19->18 20->14 20->19
Degree $20$
Order $500$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_5^3:C_4$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 125);
 
Copy content sage:G = TransitiveGroup(20, 125)
 
Copy content oscar:G = transitive_group(20, 125)
 
Copy content gap:G := TransitiveGroup(20, 125);
 

Group invariants

Abstract group:  $C_5^3:C_4$
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:  $500=2^{2} \cdot 5^{3}$
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 $n$:  $20$
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 $t$:  $125$
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);
 
$\card{\Aut(F/K)}$:  $5$
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:  $(1,4,2,5,3)(6,7,8,9,10)(16,20,19,18,17)$, $(1,17,12,6)(2,16,13,7)(3,20,14,8)(4,19,15,9)(5,18,11,10)$
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)
$2$:  $C_2$
$4$:  $C_4$
$10$:  $D_{5}$
$20$:  $F_5$ x 2, 20T2
$100$:  $C_5^2 : C_4$, 20T26 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: None

Low degree siblings

20T125 x 3

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 $2^{10}$ $25$ $2$ $10$ $( 1,14)( 2,15)( 3,11)( 4,12)( 5,13)( 6,17)( 7,16)( 8,20)( 9,19)(10,18)$
4A1 $4^{5}$ $125$ $4$ $15$ $( 1,17,14, 6)( 2,16,15, 7)( 3,20,11, 8)( 4,19,12, 9)( 5,18,13,10)$
4A-1 $4^{5}$ $125$ $4$ $15$ $( 1, 6,14,17)( 2, 7,15,16)( 3, 8,11,20)( 4, 9,12,19)( 5,10,13,18)$
5A1 $5^{4}$ $2$ $5$ $16$ $( 1, 4, 2, 5, 3)( 6, 8,10, 7, 9)(11,14,12,15,13)(16,19,17,20,18)$
5A2 $5^{4}$ $2$ $5$ $16$ $( 1, 2, 3, 4, 5)( 6,10, 9, 8, 7)(11,12,13,14,15)(16,17,18,19,20)$
5B $5^{4}$ $4$ $5$ $16$ $( 1, 3, 5, 2, 4)( 6, 7, 8, 9,10)(11,14,12,15,13)(16,17,18,19,20)$
5C $5^{4}$ $4$ $5$ $16$ $( 1, 2, 3, 4, 5)( 6, 8,10, 7, 9)(11,15,14,13,12)(16,18,20,17,19)$
5D1 $5^{4}$ $4$ $5$ $16$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,13,15,12,14)(16,19,17,20,18)$
5D2 $5^{4}$ $4$ $5$ $16$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,15,14,13,12)(16,17,18,19,20)$
5E1 $5^{2},1^{10}$ $4$ $5$ $8$ $( 6,10, 9, 8, 7)(16,20,19,18,17)$
5E2 $5^{2},1^{10}$ $4$ $5$ $8$ $( 1, 3, 5, 2, 4)(11,14,12,15,13)$
5F1 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 2, 3, 4, 5)( 6, 8,10, 7, 9)(11,13,15,12,14)$
5F-1 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(16,17,18,19,20)$
5F2 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 3, 5, 2, 4)( 6,10, 9, 8, 7)(11,15,14,13,12)$
5F-2 $5^{3},1^{5}$ $4$ $5$ $12$ $( 6, 9, 7,10, 8)(11,12,13,14,15)(16,20,19,18,17)$
5G1 $5^{3},1^{5}$ $4$ $5$ $12$ $( 6, 9, 7,10, 8)(11,14,12,15,13)(16,17,18,19,20)$
5G-1 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 2, 3, 4, 5)(11,13,15,12,14)(16,19,17,20,18)$
5G2 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 4, 2, 5, 3)(11,12,13,14,15)(16,20,19,18,17)$
5G-2 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,13,15,12,14)$
5H1 $5^{4}$ $4$ $5$ $16$ $( 1, 3, 5, 2, 4)( 6,10, 9, 8, 7)(11,12,13,14,15)(16,18,20,17,19)$
5H-1 $5^{4}$ $4$ $5$ $16$ $( 1, 5, 4, 3, 2)( 6, 8,10, 7, 9)(11,14,12,15,13)(16,20,19,18,17)$
5H2 $5^{4}$ $4$ $5$ $16$ $( 1, 5, 4, 3, 2)( 6, 9, 7,10, 8)(11,13,15,12,14)(16,20,19,18,17)$
5H-2 $5^{4}$ $4$ $5$ $16$ $( 1, 3, 5, 2, 4)( 6, 9, 7,10, 8)(11,15,14,13,12)(16,20,19,18,17)$
5I1 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 5, 4, 3, 2)(11,15,14,13,12)(16,19,17,20,18)$
5I-1 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 4, 2, 5, 3)( 6, 7, 8, 9,10)(16,20,19,18,17)$
5I2 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 4, 2, 5, 3)( 6,10, 9, 8, 7)(11,14,12,15,13)$
5I-2 $5^{3},1^{5}$ $4$ $5$ $12$ $( 6, 8,10, 7, 9)(11,12,13,14,15)(16,19,17,20,18)$
5J1 $5^{2},1^{10}$ $4$ $5$ $8$ $( 1, 2, 3, 4, 5)(16,17,18,19,20)$
5J-1 $5^{2},1^{10}$ $4$ $5$ $8$ $( 6, 7, 8, 9,10)(11,15,14,13,12)$
5J2 $5^{2},1^{10}$ $4$ $5$ $8$ $( 1, 4, 2, 5, 3)( 6, 8,10, 7, 9)$
5J-2 $5^{2},1^{10}$ $4$ $5$ $8$ $( 6, 8,10, 7, 9)(11,14,12,15,13)$
5K1 $5^{4}$ $4$ $5$ $16$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,13,15,12,14)(16,20,19,18,17)$
5K-1 $5^{4}$ $4$ $5$ $16$ $( 1, 5, 4, 3, 2)( 6, 9, 7,10, 8)(11,15,14,13,12)(16,17,18,19,20)$
5K2 $5^{4}$ $4$ $5$ $16$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,15,14,13,12)(16,19,17,20,18)$
5K-2 $5^{4}$ $4$ $5$ $16$ $( 1, 4, 2, 5, 3)( 6, 7, 8, 9,10)(11,14,12,15,13)(16,18,20,17,19)$
10A1 $10^{2}$ $50$ $10$ $18$ $( 1,15, 4,13, 2,11, 5,14, 3,12)( 6,17, 8,20,10,18, 7,16, 9,19)$
10A3 $10^{2}$ $50$ $10$ $18$ $( 1,13, 5,12, 4,11, 3,15, 2,14)( 6,20, 7,19, 8,18, 9,17,10,16)$

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

38 x 38 character table

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