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Properties

Label 25T16
25T16 1 2 1->2 12 1->12 3 2->3 6 2->6 4 3->4 5 3->5 4->5 24 4->24 5->1 18 5->18 13 6->13 21 6->21 7 22 7->22 8 8->1 23 8->23 9 9->24 25 9->25 10 19 10->19 10->25 11 14 11->14 20 11->20 12->8 16 12->16 13->2 17 13->17 14->18 14->21 15 15->19 15->20 16->14 16->15 17->9 17->15 18->3 18->11 19->12 19->22 20->13 20->16 21->8 21->11 22->9 22->10 23->4 23->10 24->6 24->23 25->7 25->17
Degree $25$
Order $150$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive yes
$p$-group no
Group: $C_5^2:S_3$

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, 16);
 
Copy content sage:G = TransitiveGroup(25, 16)
 
Copy content oscar:G = transitive_group(25, 16)
 
Copy content gap:G := TransitiveGroup(25, 16);
 

Group invariants

Abstract group:  $C_5^2:S_3$
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:  $150=2 \cdot 3 \cdot 5^{2}$
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 $t$:  $16$
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:  yes
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:  $(1,12,8)(2,6,13)(3,5,18)(4,24,23)(9,25,17)(10,19,22)(11,14,21)(15,20,16)$, $(1,2,3,4,5)(6,21,8,23,10,25,7,22,9,24)(11,20,13,17,15,19,12,16,14,18)$
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)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

15T13, 15T14, 30T37, 30T38

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 $2^{10},1^{5}$ $15$ $2$ $10$ $( 1, 9)( 2,10)( 3, 6)( 4, 7)( 5, 8)(11,22)(12,23)(13,24)(14,25)(15,21)$
3A $3^{8},1$ $50$ $3$ $16$ $( 1,12,10)( 2,17, 4)( 3,22,23)( 5, 7,11)( 8,16,24)( 9,21,18)(13,15,25)(14,20,19)$
5A1 $5^{5}$ $3$ $5$ $20$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)$
5A-1 $5^{5}$ $3$ $5$ $20$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)$
5A2 $5^{5}$ $3$ $5$ $20$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)$
5A-2 $5^{5}$ $3$ $5$ $20$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)$
5B1 $5^{5}$ $6$ $5$ $20$ $( 1,23,20,12, 9)( 2,24,16,13,10)( 3,25,17,14, 6)( 4,21,18,15, 7)( 5,22,19,11, 8)$
5B2 $5^{5}$ $6$ $5$ $20$ $( 1,12,23, 9,20)( 2,13,24,10,16)( 3,14,25, 6,17)( 4,15,21, 7,18)( 5,11,22, 8,19)$
10A1 $10^{2},5$ $15$ $10$ $22$ $( 1, 8, 4, 6, 2, 9, 5, 7, 3,10)(11,21,14,24,12,22,15,25,13,23)(16,20,19,18,17)$
10A-1 $10^{2},5$ $15$ $10$ $22$ $( 1,10, 3, 7, 5, 9, 2, 6, 4, 8)(11,23,13,25,15,22,12,24,14,21)(16,17,18,19,20)$
10A3 $10^{2},5$ $15$ $10$ $22$ $( 1, 6, 5,10, 4, 9, 3, 8, 2, 7)(11,24,15,23,14,22,13,21,12,25)(16,18,20,17,19)$
10A-3 $10^{2},5$ $15$ $10$ $22$ $( 1, 7, 2, 8, 3, 9, 4,10, 5, 6)(11,25,12,21,13,22,14,23,15,24)(16,19,17,20,18)$

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 3A 5A1 5A-1 5A2 5A-2 5B1 5B2 10A1 10A-1 10A3 10A-3
Size 1 15 50 3 3 3 3 6 6 15 15 15 15
2 P 1A 1A 3A 5A2 5A-2 5A-1 5A1 5B2 5B1 5A1 5A-1 5A-2 5A2
3 P 1A 2A 1A 5A-2 5A2 5A1 5A-1 5B2 5B1 10A3 10A-3 10A-1 10A1
5 P 1A 2A 3A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A
Type
150.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
150.5.1b R 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
150.5.2a R 2 0 −1 2 2 2 2 2 2 0 0 0 0
150.5.3a1 C 3 1 0 ζ5−2−1−ζ5−ζ52 ζ5−2+2ζ5 ζ5+2ζ52 −2ζ5−2−2−2ζ5−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5 ζ5−1 ζ5−2 ζ52
150.5.3a2 C 3 1 0 ζ5+2ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2−1−ζ5−ζ52 ζ5−2+2ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1 ζ5 ζ52 ζ5−2
150.5.3a3 C 3 1 0 −2ζ5−2−2−2ζ5−ζ52 ζ5−2−1−ζ5−ζ52 ζ5−2+2ζ5 ζ5+2ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2 ζ52 ζ5−1 ζ5
150.5.3a4 C 3 1 0 ζ5−2+2ζ5 ζ5+2ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2−1−ζ5−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ52 ζ5−2 ζ5 ζ5−1
150.5.3b1 C 3 −1 0 ζ5−2−1−ζ5−ζ52 ζ5−2+2ζ5 ζ5+2ζ52 −2ζ5−2−2−2ζ5−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5 −ζ5−1 −ζ5−2 −ζ52
150.5.3b2 C 3 −1 0 ζ5+2ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2−1−ζ5−ζ52 ζ5−2+2ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1 −ζ5 −ζ52 −ζ5−2
150.5.3b3 C 3 −1 0 −2ζ5−2−2−2ζ5−ζ52 ζ5−2−1−ζ5−ζ52 ζ5−2+2ζ5 ζ5+2ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2 −ζ52 −ζ5−1 −ζ5
150.5.3b4 C 3 −1 0 ζ5−2+2ζ5 ζ5+2ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2−1−ζ5−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ52 −ζ5−2 −ζ5 −ζ5−1
150.5.6a1 R 6 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 0 0 0 0
150.5.6a2 R 6 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 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