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Properties

Label 25T12
25T12 1 7 1->7 24 1->24 2 6 2->6 23 2->23 3 10 3->10 22 3->22 4 9 4->9 21 4->21 5 8 5->8 25 5->25 6->4 7->3 8->2 9->1 10->5 11 11->9 11->22 12 12->8 12->21 13 13->7 13->25 14 14->6 14->24 15 15->10 15->23 16 16->14 17 16->17 17->13 18 18->12 20 18->20 19 19->11 20->15 21->19 22->18 23->17 24->16 25->20
Degree $25$
Order $100$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_5^2$

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, 12);
 
Copy content sage:G = TransitiveGroup(25, 12)
 
Copy content oscar:G = transitive_group(25, 12)
 
Copy content gap:G := TransitiveGroup(25, 12);
 

Group invariants

Abstract group:  $D_5^2$
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:  $100=2^{2} \cdot 5^{2}$
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 $t$:  $12$
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:  $(1,24,16,14,6,4,21,19,11,9)(2,23,17,13,7,3,22,18,12,8)(5,25,20,15,10)$, $(1,7)(2,6)(3,10)(4,9)(5,8)(11,22)(12,21)(13,25)(14,24)(15,23)(16,17)(18,20)$
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
$4$:  $C_2^2$
$10$:  $D_{5}$ x 2
$20$:  $D_{10}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$ x 2

Low degree siblings

10T9 x 2, 20T28 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 $2^{10},1^{5}$ $5$ $2$ $10$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$
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 $2^{12},1$ $25$ $2$ $12$ $( 1, 7)( 2, 6)( 3,10)( 4, 9)( 5, 8)(11,22)(12,21)(13,25)(14,24)(15,23)(16,17)(18,20)$
5A1 $5^{5}$ $2$ $5$ $20$ $( 1,16, 6,21,11)( 2,17, 7,22,12)( 3,18, 8,23,13)( 4,19, 9,24,14)( 5,20,10,25,15)$
5A2 $5^{5}$ $2$ $5$ $20$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24)( 5,10,15,20,25)$
5B1 $5^{5}$ $2$ $5$ $20$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)$
5B2 $5^{5}$ $2$ $5$ $20$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)$
5C1 $5^{5}$ $4$ $5$ $20$ $( 1,17, 8,24,15)( 2,18, 9,25,11)( 3,19,10,21,12)( 4,20, 6,22,13)( 5,16, 7,23,14)$
5C2 $5^{5}$ $4$ $5$ $20$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22)( 5, 7,14,16,23)$
5D1 $5^{5}$ $4$ $5$ $20$ $( 1,18,10,22,14)( 2,19, 6,23,15)( 3,20, 7,24,11)( 4,16, 8,25,12)( 5,17, 9,21,13)$
5D2 $5^{5}$ $4$ $5$ $20$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23)( 5, 6,12,18,24)$
10A1 $10^{2},5$ $10$ $10$ $22$ $( 1,21,16,11, 6)( 2,25,17,15, 7, 5,22,20,12,10)( 3,24,18,14, 8, 4,23,19,13, 9)$
10A3 $10^{2},5$ $10$ $10$ $22$ $( 1,11,21, 6,16)( 2,15,22,10,17, 5,12,25, 7,20)( 3,14,23, 9,18, 4,13,24, 8,19)$
10B1 $10^{2},5$ $10$ $10$ $22$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$
10B3 $10^{2},5$ $10$ $10$ $22$ $( 1, 3, 5, 2, 4)( 6,23,10,22, 9,21, 8,25, 7,24)(11,18,15,17,14,16,13,20,12,19)$

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 5A1 5A2 5B1 5B2 5C1 5C2 5D1 5D2 10A1 10A3 10B1 10B3
Size 1 5 5 25 2 2 2 2 4 4 4 4 10 10 10 10
2 P 1A 1A 1A 1A 5A2 5A1 5B2 5B1 5C2 5C1 5D2 5D1 5A1 5A2 5B1 5B2
5 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2B 2B
Type
100.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
100.13.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
100.13.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1
100.13.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 1 1
100.13.2a1 R 2 0 2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 0
100.13.2a2 R 2 0 2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 0
100.13.2b1 R 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52
100.13.2b2 R 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5
100.13.2c1 R 2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
100.13.2c2 R 2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
100.13.2d1 R 2 0 −2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
100.13.2d2 R 2 0 −2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0
100.13.4a1 R 4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 ζ5−2+2+ζ52 −1 −ζ5−2+1−ζ52 0 0 0 0
100.13.4a2 R 4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 −ζ5−2+1−ζ52 −1 ζ5−2+2+ζ52 0 0 0 0
100.13.4b1 R 4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −ζ5−2+1−ζ52 −1 ζ5−2+2+ζ52 −1 0 0 0 0
100.13.4b2 R 4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 ζ5−2+2+ζ52 −1 −ζ5−2+1−ζ52 −1 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