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Properties

Label 40T52
40T52 1 15 1->15 33 1->33 2 16 2->16 34 2->34 3 13 3->13 36 3->36 4 14 4->14 35 4->35 5 10 5->10 21 5->21 6 9 6->9 22 6->22 7 12 7->12 24 7->24 8 11 8->11 23 8->23 9->5 9->11 10->6 10->12 11->7 11->10 12->8 12->9 13->4 37 13->37 14->3 38 14->38 15->2 39 15->39 16->1 40 16->40 17 27 17->27 17->39 18 28 18->28 18->40 19 26 19->26 19->37 20 25 20->25 20->38 21->15 21->34 22->16 22->33 23->14 23->36 24->13 24->35 25->1 30 25->30 26->2 29 26->29 27->3 32 27->32 28->4 31 28->31 29->25 29->32 30->26 30->31 31->27 31->29 32->28 32->30 33->20 33->21 34->19 34->22 35->18 35->23 36->17 36->24 37->7 37->20 38->8 38->19 39->5 39->18 40->6 40->17
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{20}:C_4$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(40, 52);
 
Copy content sage:G = TransitiveGroup(40, 52)
 
Copy content oscar:G = transitive_group(40, 52)
 
Copy content gap:G := TransitiveGroup(40, 52);
 

Group invariants

Abstract group:  $C_{20}:C_4$
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:  $80=2^{4} \cdot 5$
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 $n$:  $40$
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 $t$:  $52$
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);
 
$\card{\Aut(F/K)}$:  $8$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(40).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(40), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(40), G));
 
Generators:  $(1,15,2,16)(3,13,4,14)(5,10,6,9)(7,12,8,11)(17,39,18,40)(19,37,20,38)(21,34,22,33)(23,36,24,35)(25,30,26,29)(27,32,28,31)$, $(1,33,20,25)(2,34,19,26)(3,36,17,27)(4,35,18,28)(5,21,15,39)(6,22,16,40)(7,24,13,37)(8,23,14,38)(9,11,10,12)(29,32,30,31)$
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
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$, $C_4\times C_2$, $Q_8$
$16$:  $C_4:C_4$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $F_5$

Degree 8: $Q_8$

Degree 10: $F_5$, $F_{5}\times C_2$ x 2

Degree 20: 20T13

Low degree siblings

20T18 x 2, 40T54

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{40}$ $1$ $1$ $0$ $()$
2A $2^{20}$ $1$ $2$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
2B $2^{16},1^{8}$ $5$ $2$ $16$ $( 1,35)( 2,36)( 3,33)( 4,34)( 5,31)( 6,32)( 7,30)( 8,29)( 9,28)(10,27)(11,25)(12,26)(13,21)(14,22)(15,23)(16,24)$
2C $2^{20}$ $5$ $2$ $20$ $( 1, 2)( 3, 4)( 5,37)( 6,38)( 7,40)( 8,39)( 9,35)(10,36)(11,33)(12,34)(13,30)(14,29)(15,31)(16,32)(17,26)(18,25)(19,28)(20,27)(21,22)(23,24)$
4A $4^{10}$ $2$ $4$ $30$ $( 1,23, 2,24)( 3,21, 4,22)( 5,27, 6,28)( 7,26, 8,25)( 9,31,10,32)(11,30,12,29)(13,34,14,33)(15,36,16,35)(17,39,18,40)(19,37,20,38)$
4B $4^{10}$ $10$ $4$ $30$ $( 1,23, 2,24)( 3,21, 4,22)( 5,19, 6,20)( 7,18, 8,17)( 9,16,10,15)(11,14,12,13)(25,39,26,40)(27,37,28,38)(29,34,30,33)(31,35,32,36)$
4C1 $4^{10}$ $10$ $4$ $30$ $( 1, 4, 2, 3)( 5,14,37,29)( 6,13,38,30)( 7,15,40,31)( 8,16,39,32)( 9,26,35,17)(10,25,36,18)(11,28,33,19)(12,27,34,20)(21,24,22,23)$
4C-1 $4^{10}$ $10$ $4$ $30$ $( 1, 3, 2, 4)( 5,29,37,14)( 6,30,38,13)( 7,31,40,15)( 8,32,39,16)( 9,17,35,26)(10,18,36,25)(11,19,33,28)(12,20,34,27)(21,23,22,24)$
4D1 $4^{10}$ $10$ $4$ $30$ $( 1,29,28,39)( 2,30,27,40)( 3,31,25,37)( 4,32,26,38)( 5,17,23,12)( 6,18,24,11)( 7,19,21, 9)( 8,20,22,10)(13,36,14,35)(15,33,16,34)$
4D-1 $4^{10}$ $10$ $4$ $30$ $( 1,39,28,29)( 2,40,27,30)( 3,37,25,31)( 4,38,26,32)( 5,12,23,17)( 6,11,24,18)( 7, 9,21,19)( 8,10,22,20)(13,35,14,36)(15,34,16,33)$
5A $5^{8}$ $4$ $5$ $32$ $( 1,35,27,19,10)( 2,36,28,20, 9)( 3,33,26,18,12)( 4,34,25,17,11)( 5,38,31,24,16)( 6,37,32,23,15)( 7,39,30,22,14)( 8,40,29,21,13)$
10A $10^{4}$ $4$ $10$ $36$ $( 1,20,35, 9,27, 2,19,36,10,28)( 3,17,33,11,26, 4,18,34,12,25)( 5,23,38,15,31, 6,24,37,16,32)( 7,21,39,13,30, 8,22,40,14,29)$
20A1 $20^{2}$ $4$ $20$ $38$ $( 1,32,20, 5,35,23, 9,38,27,15, 2,31,19, 6,36,24,10,37,28,16)( 3,29,17, 7,33,21,11,39,26,13, 4,30,18, 8,34,22,12,40,25,14)$
20A-1 $20^{2}$ $4$ $20$ $38$ $( 1, 6, 9,16,19,23,28,31,35,37, 2, 5,10,15,20,24,27,32,36,38)( 3, 8,11,14,18,21,25,30,33,40, 4, 7,12,13,17,22,26,29,34,39)$

Malle's constant $a(G)$:     $1/16$

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 4A 4B 4C1 4C-1 4D1 4D-1 5A 10A 20A1 20A-1
Size 1 1 5 5 2 10 10 10 10 10 4 4 4 4
2 P 1A 1A 1A 1A 2A 2A 2C 2C 2C 2C 5A 5A 10A 10A
5 P 1A 2A 2B 2C 4A 4B 4C1 4C-1 4D1 4D-1 1A 2A 4A 4A
Type
80.31.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.31.1b R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1
80.31.1c R 1 1 1 1 −1 1 1 −1 −1 −1 1 1 −1 −1
80.31.1d R 1 1 1 1 1 −1 −1 1 −1 −1 1 1 1 1
80.31.1e1 C 1 1 −1 −1 −1 −i i 1 −i i 1 1 −1 −1
80.31.1e2 C 1 1 −1 −1 −1 i −i 1 i −i 1 1 −1 −1
80.31.1f1 C 1 1 −1 −1 1 −i i −1 i −i 1 1 1 1
80.31.1f2 C 1 1 −1 −1 1 i −i −1 −i i 1 1 1 1
80.31.2a R 2 −2 −2 2 0 0 0 0 0 0 2 −2 0 0
80.31.2b S 2 −2 2 −2 0 0 0 0 0 0 2 −2 0 0
80.31.4a R 4 4 0 0 4 0 0 0 0 0 −1 −1 −1 −1
80.31.4b R 4 4 0 0 −4 0 0 0 0 0 −1 −1 1 1
80.31.4c1 C 4 −4 0 0 0 0 0 0 0 0 −1 1 −2ζ203+ζ205−2ζ207 2ζ203−ζ205+2ζ207
80.31.4c2 C 4 −4 0 0 0 0 0 0 0 0 −1 1 2ζ203−ζ205+2ζ207 −2ζ203+ζ205−2ζ207

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