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Properties

Label 40T54
40T54 1 34 1->34 40 1->40 2 33 2->33 39 2->39 3 36 3->36 37 3->37 4 35 4->35 38 4->38 5 11 5->11 21 5->21 6 12 6->12 22 6->22 7 9 7->9 24 7->24 8 10 8->10 23 8->23 9->11 9->22 10->12 10->21 11->23 12->24 13 13->36 13->38 14 14->35 14->37 15 15->33 15->40 16 16->34 16->39 17 17->5 28 17->28 18 18->6 27 18->27 19 19->7 26 19->26 20 20->8 25 20->25 21->16 21->20 22->15 22->19 23->14 23->17 24->13 24->18 25->2 32 25->32 26->1 31 26->31 27->4 29 27->29 28->3 30 28->30 29->2 29->31 30->1 30->32 31->3 32->4 33->20 34->19 35->18 36->17 37->8 37->26 38->7 38->25 39->5 39->27 40->6 40->28
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{20}:C_4$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(40, 54);
 
Copy content sage:G = TransitiveGroup(40, 54)
 
Copy content oscar:G = transitive_group(40, 54)
 
Copy content gap:G := TransitiveGroup(40, 54);
 

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 $t$:  $54$
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:  $(1,34,19,26)(2,33,20,25)(3,36,17,28)(4,35,18,27)(5,21,16,39)(6,22,15,40)(7,24,13,38)(8,23,14,37)(9,11)(10,12)(29,31)(30,32)$, $(1,40,28,30)(2,39,27,29)(3,37,26,31)(4,38,25,32)(5,11,23,17)(6,12,24,18)(7,9,22,19)(8,10,21,20)(13,36)(14,35)(15,33)(16,34)$
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: $C_2^2$, $D_{4}$ x 2

Degree 5: $F_5$

Degree 8: $D_4$

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

Degree 20: 20T13, 20T18 x 2

Low degree siblings

20T18 x 2, 40T52

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

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