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Properties

Label 25T11
25T11 1 4 1->4 10 1->10 2 5 2->5 9 2->9 3 3->1 8 3->8 4->2 7 4->7 5->3 6 5->6 22 6->22 25 6->25 23 7->23 24 7->24 8->23 8->24 9->22 9->25 21 10->21 10->21 11 14 11->14 20 11->20 12 13 12->13 16 12->16 17 13->17 18 14->18 15 19 15->19 16->2 16->13 17->1 17->14 18->5 18->15 19->4 19->11 20->3 20->12 21->6 21->17 22->7 22->16 23->8 23->20 24->9 24->19 25->10 25->18
Degree $25$
Order $100$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_5:F_5$

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, 11);
 
Copy content sage:G = TransitiveGroup(25, 11)
 
Copy content oscar:G = transitive_group(25, 11)
 
Copy content gap:G := TransitiveGroup(25, 11);
 

Group invariants

Abstract group:  $C_5:F_5$
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:  $100=2^{2} \cdot 5^{2}$
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 $t$:  $11$
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:  $(1,10,21,17)(2,9,22,16)(3,8,23,20)(4,7,24,19)(5,6,25,18)(11,14)(12,13)$, $(1,4,2,5,3)(6,22,7,23,8,24,9,25,10,21)(11,20,12,16,13,17,14,18,15,19)$
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)
$2$:  $C_2$
$4$:  $C_4$
$10$:  $D_{5}$
$20$:  $F_5$, 20T2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$, $F_5$

Low degree siblings

20T26

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 $2^{10},1^{5}$ $5$ $2$ $10$ $( 6,24)( 7,25)( 8,21)( 9,22)(10,23)(11,17)(12,18)(13,19)(14,20)(15,16)$
4A1 $4^{5},2^{2},1$ $25$ $4$ $17$ $( 1,12, 8,24)( 2,11, 9,23)( 3,15,10,22)( 4,14, 6,21)( 5,13, 7,25)(16,17)(18,20)$
4A-1 $4^{5},2^{2},1$ $25$ $4$ $17$ $( 1,24, 8,12)( 2,23, 9,11)( 3,22,10,15)( 4,21, 6,14)( 5,25, 7,13)(16,17)(18,20)$
5A1 $5^{5}$ $2$ $5$ $20$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)$
5A2 $5^{5}$ $2$ $5$ $20$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)$
5B $5^{5}$ $4$ $5$ $20$ $( 1,20, 8,21,14)( 2,16, 9,22,15)( 3,17,10,23,11)( 4,18, 6,24,12)( 5,19, 7,25,13)$
5C1 $5^{5}$ $4$ $5$ $20$ $( 1,17, 7,22,12)( 2,18, 8,23,13)( 3,19, 9,24,14)( 4,20,10,25,15)( 5,16, 6,21,11)$
5C-1 $5^{5}$ $4$ $5$ $20$ $( 1,10,13,16,24)( 2, 6,14,17,25)( 3, 7,15,18,21)( 4, 8,11,19,22)( 5, 9,12,20,23)$
5C2 $5^{5}$ $4$ $5$ $20$ $( 1,16,10,24,13)( 2,17, 6,25,14)( 3,18, 7,21,15)( 4,19, 8,22,11)( 5,20, 9,23,12)$
5C-2 $5^{5}$ $4$ $5$ $20$ $( 1,19, 6,23,15)( 2,20, 7,24,11)( 3,16, 8,25,12)( 4,17, 9,21,13)( 5,18,10,22,14)$
10A1 $10^{2},5$ $10$ $10$ $22$ $( 1, 3, 5, 2, 4)( 6,21,10,25, 9,24, 8,23, 7,22)(11,19,15,18,14,17,13,16,12,20)$
10A3 $10^{2},5$ $10$ $10$ $22$ $( 1, 5, 4, 3, 2)( 6,23, 9,21, 7,24,10,22, 8,25)(11,16,14,19,12,17,15,20,13,18)$

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 4A1 4A-1 5A1 5A2 5B 5C1 5C-1 5C2 5C-2 10A1 10A3
Size 1 5 25 25 2 2 4 4 4 4 4 10 10
2 P 1A 1A 2A 2A 5A2 5A1 5B 5C2 5C-2 5C-1 5C1 5A1 5A2
5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 1A 1A 1A 2A 2A
Type
100.10.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
100.10.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1
100.10.1c1 C 1 −1 −i i 1 1 1 1 1 1 1 −1 −1
100.10.1c2 C 1 −1 i −i 1 1 1 1 1 1 1 −1 −1
100.10.2a1 R 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
100.10.2a2 R 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
100.10.2b1 S 2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 2 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
100.10.2b2 S 2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 2 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
100.10.4a R 4 0 0 0 4 4 −1 −1 −1 −1 −1 0 0
100.10.4b1 C 4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 ζ5−2−ζ5−ζ52 ζ5−2+1+2ζ5 −1 −2ζ5−2−1−2ζ5−ζ52 1+ζ5+2ζ52 0 0
100.10.4b2 C 4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 1+ζ5+2ζ52 −2ζ5−2−1−2ζ5−ζ52 −1 ζ5−2+1+2ζ5 ζ5−2−ζ5−ζ52 0 0
100.10.4b3 C 4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −2ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ5−ζ52 −1 1+ζ5+2ζ52 ζ5−2+1+2ζ5 0 0
100.10.4b4 C 4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+1+2ζ5 1+ζ5+2ζ52 −1 ζ5−2−ζ5−ζ52 −2ζ5−2−1−2ζ5−ζ52 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