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Properties

Label 25T19
25T19 1 12 1->12 22 1->22 2 14 2->14 21 2->21 3 11 3->11 25 3->25 4 13 4->13 24 4->24 5 15 5->15 23 5->23 6 6->2 6->2 7 7->1 7->4 8 8->1 8->5 9 9->3 9->4 10 10->3 10->5 11->7 17 11->17 12->6 19 12->19 13->10 16 13->16 14->9 18 14->18 15->8 20 15->20 16->7 16->12 17->9 17->11 18->6 18->15 19->8 19->14 20->10 20->13 21->17 21->22 22->16 22->24 23->20 23->21 24->19 24->23 25->18
Degree $25$
Order $200$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_5:F_5$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(25, 19);
 
Copy content sage:G = TransitiveGroup(25, 19)
 
Copy content oscar:G = transitive_group(25, 19)
 
Copy content gap:G := TransitiveGroup(25, 19);
 

Group invariants

Abstract group:  $D_5:F_5$
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:  $200=2^{3} \cdot 5^{2}$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
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 $n$:  $25$
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 $t$:  $19$
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);
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(25).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(25), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(25), G));
 
Generators:  $(1,22,16,12,6,2,21,17,11,7)(3,25,18,15,8,5,23,20,13,10)(4,24,19,14,9)$, $(1,12,19,8)(2,14,18,6)(3,11,17,9)(4,13,16,7)(5,15,20,10)(21,22,24,23)$
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, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$ x 2
$40$:  $F_{5}\times C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$ x 2

Low degree siblings

10T17 x 2, 20T54 x 2, 40T169 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{25}$ $1$ $1$ $0$ $()$
2A $2^{10},1^{5}$ $5$ $2$ $10$ $( 1, 5)( 2, 4)( 6,10)( 7, 9)(11,15)(12,14)(16,20)(17,19)(21,25)(22,24)$
2B $2^{10},1^{5}$ $5$ $2$ $10$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$
2C $2^{12},1$ $25$ $2$ $12$ $( 1, 4)( 2, 3)( 6,24)( 7,23)( 8,22)( 9,21)(10,25)(11,19)(12,18)(13,17)(14,16)(15,20)$
4A1 $4^{6},1$ $25$ $4$ $18$ $( 1, 2, 4, 3)( 6,12,24,18)( 7,14,23,16)( 8,11,22,19)( 9,13,21,17)(10,15,25,20)$
4A-1 $4^{6},1$ $25$ $4$ $18$ $( 1, 3, 4, 2)( 6,18,24,12)( 7,16,23,14)( 8,19,22,11)( 9,17,21,13)(10,20,25,15)$
4B1 $4^{6},1$ $25$ $4$ $18$ $( 1, 5, 3, 4)( 6,20,23,14)( 7,17,22,12)( 8,19,21,15)( 9,16,25,13)(10,18,24,11)$
4B-1 $4^{6},1$ $25$ $4$ $18$ $( 1, 4, 3, 5)( 6,14,23,20)( 7,12,22,17)( 8,15,21,19)( 9,13,25,16)(10,11,24,18)$
5A $5^{5}$ $4$ $5$ $20$ $( 1,21,16,11, 6)( 2,22,17,12, 7)( 3,23,18,13, 8)( 4,24,19,14, 9)( 5,25,20,15,10)$
5B $5^{5}$ $4$ $5$ $20$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)$
5C $5^{5}$ $8$ $5$ $20$ $( 1,24,17,15, 8)( 2,25,18,11, 9)( 3,21,19,12,10)( 4,22,20,13, 6)( 5,23,16,14, 7)$
5D $5^{5}$ $8$ $5$ $20$ $( 1,22,18,14,10)( 2,23,19,15, 6)( 3,24,20,11, 7)( 4,25,16,12, 8)( 5,21,17,13, 9)$
10A $10^{2},5$ $20$ $10$ $22$ $( 1,15,21,10,16, 5,11,25, 6,20)( 2,14,22, 9,17, 4,12,24, 7,19)( 3,13,23, 8,18)$
10B $10^{2},5$ $20$ $10$ $22$ $( 1, 5, 4, 3, 2)( 6,25, 9,23, 7,21,10,24, 8,22)(11,20,14,18,12,16,15,19,13,17)$

Malle's constant $a(G)$:     $1/10$

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 4A1 4A-1 4B1 4B-1 5A 5B 5C 5D 10A 10B
Size 1 5 5 25 25 25 25 25 4 4 8 8 20 20
2 P 1A 1A 1A 1A 2C 2C 2C 2C 5A 5B 5C 5D 5A 5B
5 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 1A 1A 1A 1A 2A 2B
Type
200.42.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
200.42.1b R 1 −1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1
200.42.1c R 1 −1 −1 1 1 1 −1 −1 1 1 1 1 −1 −1
200.42.1d R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1
200.42.1e1 C 1 −1 1 −1 −i i −i i 1 1 1 1 −1 1
200.42.1e2 C 1 −1 1 −1 i −i i −i 1 1 1 1 −1 1
200.42.1f1 C 1 1 −1 −1 −i i i −i 1 1 1 1 1 −1
200.42.1f2 C 1 1 −1 −1 i −i −i i 1 1 1 1 1 −1
200.42.4a R 4 0 4 0 0 0 0 0 4 −1 −1 −1 0 −1
200.42.4b R 4 4 0 0 0 0 0 0 −1 4 −1 −1 −1 0
200.42.4c R 4 −4 0 0 0 0 0 0 −1 4 −1 −1 1 0
200.42.4d R 4 0 −4 0 0 0 0 0 4 −1 −1 −1 0 1
200.42.8a R 8 0 0 0 0 0 0 0 −2 −2 −2 3 0 0
200.42.8b R 8 0 0 0 0 0 0 0 −2 −2 3 −2 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