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Properties

Label 25T4
25T4 1 17 1->17 24 1->24 2 16 2->16 23 2->23 3 20 3->20 22 3->22 4 19 4->19 21 4->21 5 18 5->18 25 5->25 6 12 6->12 6->17 7 11 7->11 7->16 8 15 8->15 8->20 9 14 9->14 9->19 10 13 10->13 10->18 11->12 13->15 21->22 23->25
Degree $25$
Order $50$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{25}$

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Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(25, 4);
 
Copy content sage:G = TransitiveGroup(25, 4)
 
Copy content oscar:G = transitive_group(25, 4)
 

Group invariants

Abstract group:  $D_{25}$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $50=2 \cdot 5^{2}$
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 $n$:  $25$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Transitive number $t$:  $4$
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)
 
$\card{\Aut(F/K)}$:  $1$
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])
 
Generators:  $(1,17)(2,16)(3,20)(4,19)(5,18)(6,12)(7,11)(8,15)(9,14)(10,13)(21,22)(23,25)$, $(1,24)(2,23)(3,22)(4,21)(5,25)(6,17)(7,16)(8,20)(9,19)(10,18)(11,12)(13,15)$
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)
$2$:  $C_2$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Low degree siblings

There are no siblings 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 $2^{12},1$ $25$ $2$ $12$ $( 2, 5)( 3, 4)( 6,24)( 7,23)( 8,22)( 9,21)(10,25)(11,17)(12,16)(13,20)(14,19)(15,18)$
5A1 $5^{5}$ $2$ $5$ $20$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)$
5A2 $5^{5}$ $2$ $5$ $20$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)$
25A1 $25$ $2$ $25$ $24$ $( 1,23,20,14, 8, 2,24,16,15, 9, 3,25,17,11,10, 4,21,18,12, 6, 5,22,19,13, 7)$
25A2 $25$ $2$ $25$ $24$ $( 1,20, 8,24,15, 3,17,10,21,12, 5,19, 7,23,14, 2,16, 9,25,11, 4,18, 6,22,13)$
25A3 $25$ $2$ $25$ $24$ $( 1,14,24, 9,17, 4,12,22, 7,20, 2,15,25,10,18, 5,13,23, 8,16, 3,11,21, 6,19)$
25A4 $25$ $2$ $25$ $24$ $( 1, 8,15,17,21, 5, 7,14,16,25, 4, 6,13,20,24, 3,10,12,19,23, 2, 9,11,18,22)$
25A6 $25$ $2$ $25$ $24$ $( 1,24,17,12, 7, 2,25,18,13, 8, 3,21,19,14, 9, 4,22,20,15,10, 5,23,16,11, 6)$
25A7 $25$ $2$ $25$ $24$ $( 1,16,10,22,14, 3,18, 7,24,11, 5,20, 9,21,13, 2,17, 6,23,15, 4,19, 8,25,12)$
25A8 $25$ $2$ $25$ $24$ $( 1,15,21, 7,16, 4,13,24,10,19, 2,11,22, 8,17, 5,14,25, 6,20, 3,12,23, 9,18)$
25A9 $25$ $2$ $25$ $24$ $( 1,21,16,13,10, 2,22,17,14, 6, 3,23,18,15, 7, 4,24,19,11, 8, 5,25,20,12, 9)$
25A11 $25$ $2$ $25$ $24$ $( 1,25,19,15, 6, 2,21,20,11, 7, 3,22,16,12, 8, 4,23,17,13, 9, 5,24,18,14,10)$
25A12 $25$ $2$ $25$ $24$ $( 1,17, 7,25,13, 3,19, 9,22,15, 5,16, 6,24,12, 2,18, 8,21,14, 4,20,10,23,11)$

Malle's constant $a(G)$:     $1/12$

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 5A1 5A2 25A1 25A2 25A3 25A4 25A6 25A7 25A8 25A9 25A11 25A12
Size 1 25 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 5A2 5A1 25A2 25A4 25A6 25A8 25A12 25A11 25A9 25A7 25A3 25A1
5 P 1A 2A 1A 1A 5A1 5A2 5A2 5A1 5A1 5A2 5A2 5A1 5A1 5A2
Type
50.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
50.1.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1
50.1.2a1 R 2 0 2 2 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
50.1.2a2 R 2 0 2 2 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
50.1.2b1 R 2 0 ζ25−10+ζ2510 ζ25−5+ζ255 ζ25−11+ζ2511 ζ25−4+ζ254 ζ25−6+ζ256 ζ25−3+ζ253 ζ25−2+ζ252 ζ25−1+ζ25 ζ25−7+ζ257 ζ25−9+ζ259 ζ25−12+ζ2512 ζ25−8+ζ258
50.1.2b2 R 2 0 ζ25−10+ζ2510 ζ25−5+ζ255 ζ25−9+ζ259 ζ25−1+ζ25 ζ25−11+ζ2511 ζ25−7+ζ257 ζ25−12+ζ2512 ζ25−6+ζ256 ζ25−8+ζ258 ζ25−4+ζ254 ζ25−3+ζ253 ζ25−2+ζ252
50.1.2b3 R 2 0 ζ25−10+ζ2510 ζ25−5+ζ255 ζ25−6+ζ256 ζ25−9+ζ259 ζ25−1+ζ25 ζ25−12+ζ2512 ζ25−8+ζ258 ζ25−4+ζ254 ζ25−3+ζ253 ζ25−11+ζ2511 ζ25−2+ζ252 ζ25−7+ζ257
50.1.2b4 R 2 0 ζ25−10+ζ2510 ζ25−5+ζ255 ζ25−4+ζ254 ζ25−6+ζ256 ζ25−9+ζ259 ζ25−8+ζ258 ζ25−3+ζ253 ζ25−11+ζ2511 ζ25−2+ζ252 ζ25−1+ζ25 ζ25−7+ζ257 ζ25−12+ζ2512
50.1.2b5 R 2 0 ζ25−10+ζ2510 ζ25−5+ζ255 ζ25−1+ζ25 ζ25−11+ζ2511 ζ25−4+ζ254 ζ25−2+ζ252 ζ25−7+ζ257 ζ25−9+ζ259 ζ25−12+ζ2512 ζ25−6+ζ256 ζ25−8+ζ258 ζ25−3+ζ253
50.1.2b6 R 2 0 ζ25−5+ζ255 ζ25−10+ζ2510 ζ25−12+ζ2512 ζ25−7+ζ257 ζ25−2+ζ252 ζ25−1+ζ25 ζ25−9+ζ259 ζ25−8+ζ258 ζ25−6+ζ256 ζ25−3+ζ253 ζ25−4+ζ254 ζ25−11+ζ2511
50.1.2b7 R 2 0 ζ25−5+ζ255 ζ25−10+ζ2510 ζ25−8+ζ258 ζ25−12+ζ2512 ζ25−7+ζ257 ζ25−9+ζ259 ζ25−6+ζ256 ζ25−3+ζ253 ζ25−4+ζ254 ζ25−2+ζ252 ζ25−11+ζ2511 ζ25−1+ζ25
50.1.2b8 R 2 0 ζ25−5+ζ255 ζ25−10+ζ2510 ζ25−7+ζ257 ζ25−2+ζ252 ζ25−3+ζ253 ζ25−11+ζ2511 ζ25−1+ζ25 ζ25−12+ζ2512 ζ25−9+ζ259 ζ25−8+ζ258 ζ25−6+ζ256 ζ25−4+ζ254
50.1.2b9 R 2 0 ζ25−5+ζ255 ζ25−10+ζ2510 ζ25−3+ζ253 ζ25−8+ζ258 ζ25−12+ζ2512 ζ25−6+ζ256 ζ25−4+ζ254 ζ25−2+ζ252 ζ25−11+ζ2511 ζ25−7+ζ257 ζ25−1+ζ25 ζ25−9+ζ259
50.1.2b10 R 2 0 ζ25−5+ζ255 ζ25−10+ζ2510 ζ25−2+ζ252 ζ25−3+ζ253 ζ25−8+ζ258 ζ25−4+ζ254 ζ25−11+ζ2511 ζ25−7+ζ257 ζ25−1+ζ25 ζ25−12+ζ2512 ζ25−9+ζ259 ζ25−6+ζ256

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