Properties

Label 20T115
20T115 1 8 1->8 18 1->18 2 7 2->7 17 2->17 3 10 3->10 20 3->20 4 9 4->9 19 4->19 5 5->1 12 5->12 6 6->2 11 6->11 16 7->16 7->18 15 8->15 8->17 9->6 9->15 10->5 10->16 11->12 13 13->9 13->19 14 14->10 14->20 15->14 16->13 17->3 17->14 18->4 18->13 19->1 20->2
Degree $20$
Order $400$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_5^2:\OD_{16}$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 115);
 
Copy content sage:G = TransitiveGroup(20, 115)
 
Copy content oscar:G = transitive_group(20, 115)
 
Copy content gap:G := TransitiveGroup(20, 115);
 

Group invariants

Abstract group:  $C_5^2:\OD_{16}$
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:  $400=2^{4} \cdot 5^{2}$
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 $t$:  $115$
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);
 
$\card{\Aut(F/K)}$:  $2$
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:  $(1,8,17,3,10,16,13,19)(2,7,18,4,9,15,14,20)(5,12)(6,11)$, $(1,18,13,9,6,2,17,14,10,5)(3,20)(4,19)(7,16)(8,15)(11,12)$
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 2, $C_2^2$
$8$:  $C_4\times C_2$
$16$:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 10: $(C_5^2 : C_8):C_2$

Low degree siblings

10T28, 20T104, 20T107, 20T109, 25T31, 40T397, 40T398, 40T399, 40T400

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 $2^{10}$ $10$ $2$ $10$ $( 1, 2)( 3, 4)( 5,17)( 6,18)( 7, 8)( 9,13)(10,14)(11,12)(15,16)(19,20)$
2B $2^{8},1^{4}$ $25$ $2$ $8$ $( 1,10)( 2, 9)( 3,16)( 4,15)( 7,12)( 8,11)(13,17)(14,18)$
4A1 $4^{4},1^{4}$ $25$ $4$ $12$ $( 1,13,10,17)( 2,14, 9,18)( 3, 8,19,16)( 4, 7,20,15)$
4A-1 $4^{4},1^{4}$ $25$ $4$ $12$ $( 1,17,10,13)( 2,18, 9,14)( 3,16,19, 8)( 4,15,20, 7)$
4B $4^{4},2^{2}$ $50$ $4$ $14$ $( 1,18,10,14)( 2,17, 9,13)( 3,12,16, 7)( 4,11,15, 8)( 5, 6)(19,20)$
5A $5^{2},1^{10}$ $8$ $5$ $8$ $( 3,16, 8,19,11)( 4,15, 7,20,12)$
5B $5^{4}$ $16$ $5$ $16$ $( 1,17,13,10, 6)( 2,18,14, 9, 5)( 3,11,19, 8,16)( 4,12,20, 7,15)$
8A1 $8^{2},2^{2}$ $50$ $8$ $16$ $( 1, 7,13,20,10,15,17, 4)( 2, 8,14,19, 9,16,18, 3)( 5,11)( 6,12)$
8A-1 $8^{2},2^{2}$ $50$ $8$ $16$ $( 1, 4,17,15,10,20,13, 7)( 2, 3,18,16, 9,19,14, 8)( 5,11)( 6,12)$
8B1 $8^{2},2^{2}$ $50$ $8$ $16$ $( 1,16)( 2,15)( 3,13,19,17, 8,10,11, 6)( 4,14,20,18, 7, 9,12, 5)$
8B-1 $8^{2},2^{2}$ $50$ $8$ $16$ $( 1,16)( 2,15)( 3, 6,11,10, 8,17,19,13)( 4, 5,12, 9, 7,18,20,14)$
10A $10,2^{5}$ $40$ $10$ $14$ $( 1, 2)( 3,20,16,12, 8, 4,19,15,11, 7)( 5,17)( 6,18)( 9,13)(10,14)$

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 4A1 4A-1 4B 5A 5B 8A1 8A-1 8B1 8B-1 10A
Size 1 10 25 25 25 50 8 16 50 50 50 50 40
2 P 1A 1A 1A 2B 2B 2B 5A 5B 4A1 4A-1 4A1 4A-1 5A
5 P 1A 2A 2B 4A1 4A-1 4B 1A 1A 8A1 8A-1 8B1 8B-1 2A
Type
400.206.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
400.206.1b R 1 −1 1 1 1 −1 1 1 −1 1 1 −1 −1
400.206.1c R 1 −1 1 1 1 −1 1 1 1 −1 −1 1 −1
400.206.1d R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1
400.206.1e1 C 1 −1 1 −1 −1 1 1 1 −i −i i i −1
400.206.1e2 C 1 −1 1 −1 −1 1 1 1 i i −i −i −1
400.206.1f1 C 1 1 1 −1 −1 −1 1 1 −i i −i i 1
400.206.1f2 C 1 1 1 −1 −1 −1 1 1 i −i i −i 1
400.206.2a1 C 2 0 −2 −2i 2i 0 2 2 0 0 0 0 0
400.206.2a2 C 2 0 −2 2i −2i 0 2 2 0 0 0 0 0
400.206.8a R 8 4 0 0 0 0 3 −2 0 0 0 0 −1
400.206.8b R 8 −4 0 0 0 0 3 −2 0 0 0 0 1
400.206.16a R 16 0 0 0 0 0 −4 1 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