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Properties

Label 25T38
25T38 1 4 1->4 14 1->14 2 5 2->5 11 2->11 3 3->1 13 3->13 4->2 15 4->15 5->3 12 5->12 6 24 6->24 25 6->25 7 7->24 7->25 8 21 8->21 23 8->23 9 22 9->22 9->22 10 10->21 10->23 11->9 18 11->18 12->6 17 12->17 13->8 16 13->16 14->10 20 14->20 15->7 19 15->19 16->1 16->11 17->3 17->15 18->5 18->14 19->2 19->13 20->4 20->12 21->9 21->19 22->8 22->17 23->7 23->20 24->6 24->18 25->10 25->16
Degree $25$
Order $500$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_5^2:D_{10}$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(25, 38);
 
Copy content sage:G = TransitiveGroup(25, 38)
 
Copy content oscar:G = transitive_group(25, 38)
 
Copy content gap:G := TransitiveGroup(25, 38);
 

Group invariants

Abstract group:  $C_5^2:D_{10}$
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 $n$:  $25$
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 $t$:  $38$
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:  $(1,4,2,5,3)(6,25,10,21,9,22,8,23,7,24)(11,18,14,20,12,17,15,19,13,16)$, $(1,14,10,23,20,4,15,7,25,16)(2,11,9,22,17,3,13,8,21,19)(5,12,6,24,18)$
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
$100$:  $D_5^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Low degree siblings

25T38

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 $2^{10},1^{5}$ $25$ $2$ $10$ $( 1,22)( 2,23)( 3,24)( 4,25)( 5,21)( 6,15)( 7,12)( 8,14)( 9,11)(10,13)$
2B $2^{12},1$ $25$ $2$ $12$ $( 1,24)( 2,23)( 3,22)( 4,21)( 5,25)( 6,15)( 7,13)( 8,11)( 9,14)(10,12)(16,17)(18,20)$
2C $2^{10},1^{5}$ $25$ $2$ $10$ $( 1, 3)( 4, 5)( 6, 8)( 9,10)(12,15)(13,14)(16,19)(17,18)(21,22)(23,25)$
5A1 $5^{5}$ $2$ $5$ $20$ $( 1, 5, 4, 3, 2)( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22)$
5A2 $5^{5}$ $2$ $5$ $20$ $( 1, 4, 2, 5, 3)( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)(21,24,22,25,23)$
5B1 $5^{4},1^{5}$ $10$ $5$ $16$ $( 1, 2, 3, 4, 5)( 6, 8,10, 7, 9)(11,12,13,14,15)(21,25,24,23,22)$
5B2 $5^{4},1^{5}$ $10$ $5$ $16$ $( 1, 3, 5, 2, 4)( 6,10, 9, 8, 7)(11,13,15,12,14)(21,24,22,25,23)$
5C1 $5^{5}$ $10$ $5$ $20$ $( 1, 8,18,13,23)( 2, 7,20,11,24)( 3, 6,17,14,25)( 4,10,19,12,21)( 5, 9,16,15,22)$
5C2 $5^{5}$ $10$ $5$ $20$ $( 1,18,23, 8,13)( 2,20,24, 7,11)( 3,17,25, 6,14)( 4,19,21,10,12)( 5,16,22, 9,15)$
5D1 $5^{5}$ $20$ $5$ $20$ $( 1,20,21, 7,13)( 2,17,22, 6,11)( 3,19,23,10,14)( 4,16,24, 9,12)( 5,18,25, 8,15)$
5D2 $5^{5}$ $20$ $5$ $20$ $( 1,24,11,18,10)( 2,25,14,20, 9)( 3,21,12,17, 8)( 4,22,15,19, 7)( 5,23,13,16, 6)$
5E1 $5^{5}$ $20$ $5$ $20$ $( 1,20,22, 9,11)( 2,17,23, 8,14)( 3,19,24, 7,12)( 4,16,25, 6,15)( 5,18,21,10,13)$
5E2 $5^{5}$ $20$ $5$ $20$ $( 1,23,14,20, 6)( 2,24,12,17,10)( 3,25,15,19, 9)( 4,21,13,16, 8)( 5,22,11,18, 7)$
10A1 $10^{2},5$ $50$ $10$ $22$ $( 1,24, 5,23, 4,22, 3,21, 2,25)( 6,11, 7,13, 8,15, 9,12,10,14)(16,20,19,18,17)$
10A3 $10^{2},5$ $50$ $10$ $22$ $( 1,23, 3,25, 5,22, 2,24, 4,21)( 6,13, 9,14, 7,15,10,11, 8,12)(16,18,20,17,19)$
10B1 $10^{2},2^{2},1$ $50$ $10$ $20$ $( 1,21, 2,25, 3,24, 4,23, 5,22)( 6,13, 8,14,10,15, 7,11, 9,12)(16,17)(18,20)$
10B3 $10^{2},2^{2},1$ $50$ $10$ $20$ $( 1,25, 4,22, 2,24, 5,21, 3,23)( 6,14, 7,12, 8,15, 9,13,10,11)(16,17)(18,20)$
10C1 $10^{2},5$ $50$ $10$ $22$ $( 1,14, 8,25,18, 3,13, 6,23,17)( 2,11, 7,24,20)( 4,15,10,22,19, 5,12, 9,21,16)$
10C3 $10^{2},5$ $50$ $10$ $22$ $( 1,25,13,17, 8, 3,23,14,18, 6)( 2,24,11,20, 7)( 4,22,12,16,10, 5,21,15,19, 9)$

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 5E1 5E2 10A1 10A3 10B1 10B3 10C1 10C3
Size 1 25 25 25 2 2 10 10 10 10 20 20 20 20 50 50 50 50 50 50
2 P 1A 1A 1A 1A 5A2 5A1 5B2 5B1 5C2 5C1 5D2 5D1 5E2 5E1 5A1 5A2 5B1 5B2 5C1 5C2
5 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2B 2B 2C 2C
Type
500.27.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
500.27.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 1 1 −1
500.27.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 −1 1 1 −1 −1 −1
500.27.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1
500.27.2a1 R 2 0 0 2 2 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0
500.27.2a2 R 2 0 0 2 2 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0
500.27.2b1 R 2 0 2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0
500.27.2b2 R 2 0 2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0
500.27.2c1 R 2 0 −2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0
500.27.2c2 R 2 0 −2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0
500.27.2d1 R 2 0 0 −2 2 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0
500.27.2d2 R 2 0 0 −2 2 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0
500.27.4a1 R 4 0 0 0 4 4 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 −1 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 0 0 0 0 0 0
500.27.4a2 R 4 0 0 0 4 4 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 −1 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 0 0 0 0 0 0
500.27.4b1 R 4 0 0 0 4 4 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 −1 0 0 0 0 0 0
500.27.4b2 R 4 0 0 0 4 4 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 −1 0 0 0 0 0 0
500.27.10a1 R 10 2 0 0 5ζ5−2+5ζ52 5ζ5−1+5ζ5 0 0 0 0 0 0 0 0 ζ5−1+ζ5 0 0 0 0 ζ5−2+ζ52
500.27.10a2 R 10 2 0 0 5ζ5−1+5ζ5 5ζ5−2+5ζ52 0 0 0 0 0 0 0 0 ζ5−2+ζ52 0 0 0 0 ζ5−1+ζ5
500.27.10b1 R 10 −2 0 0 5ζ5−2+5ζ52 5ζ5−1+5ζ5 0 0 0 0 0 0 0 0 −ζ5−1−ζ5 0 0 0 0 −ζ5−2−ζ52
500.27.10b2 R 10 −2 0 0 5ζ5−1+5ζ5 5ζ5−2+5ζ52 0 0 0 0 0 0 0 0 −ζ5−2−ζ52 0 0 0 0 −ζ5−1−ζ5

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