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Properties

Label 40T9
40T9 1 25 1->25 37 1->37 2 26 2->26 38 2->38 3 28 3->28 40 3->40 4 27 4->27 39 4->39 5 5->3 24 5->24 6 6->4 23 6->23 7 7->1 21 7->21 8 8->2 22 8->22 9 9->6 19 9->19 10 10->5 20 10->20 11 11->7 17 11->17 12 12->8 18 12->18 13 13->10 16 13->16 14 14->9 15 14->15 15->12 16->11 17->14 18->13 19->16 20->15 21->18 22->17 23->20 24->19 25->22 26->21 27->24 28->23 29 29->25 29->38 30 30->26 30->37 31 31->27 31->40 32 32->28 32->39 33 33->29 35 33->35 34 34->30 36 34->36 35->32 36->31 37->35 38->36 39->34 40->33
Degree $40$
Order $40$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_4\times D_5$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(40, 9);
 
Copy content sage:G = TransitiveGroup(40, 9)
 
Copy content oscar:G = transitive_group(40, 9)
 
Copy content gap:G := TransitiveGroup(40, 9);
 

Group invariants

Abstract group:  $C_4\times D_5$
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:  $40=2^{3} \cdot 5$
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 $t$:  $9$
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);
 
$\card{\Aut(F/K)}$:  $40$
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:  $(1,25)(2,26)(3,28)(4,27)(5,24)(6,23)(7,21)(8,22)(9,19)(10,20)(11,17)(12,18)(13,16)(14,15)(29,38)(30,37)(31,40)(32,39)(33,35)(34,36)$, $(1,37,35,32,28,23,20,15,12,8,2,38,36,31,27,24,19,16,11,7)(3,40,33,29,25,22,17,14,9,6,4,39,34,30,26,21,18,13,10,5)$
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, $C_2^2$
$8$:  $C_4\times C_2$
$10$:  $D_{5}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 5: $D_{5}$

Degree 8: $C_4\times C_2$

Degree 10: $D_5$, $D_{10}$ x 2

Degree 20: 20T4, 20T6 x 2

Low degree siblings

20T6 x 2

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 $2^{20}$ $5$ $2$ $20$ $( 1,25)( 2,26)( 3,28)( 4,27)( 5,24)( 6,23)( 7,21)( 8,22)( 9,19)(10,20)(11,17)(12,18)(13,16)(14,15)(29,38)(30,37)(31,40)(32,39)(33,35)(34,36)$
2C $2^{20}$ $5$ $2$ $20$ $( 1,17)( 2,18)( 3,20)( 4,19)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)(21,37)(22,38)(23,39)(24,40)(25,35)(26,36)(27,34)(28,33)(29,31)(30,32)$
4A1 $4^{10}$ $1$ $4$ $30$ $( 1,23, 2,24)( 3,22, 4,21)( 5,25, 6,26)( 7,28, 8,27)( 9,30,10,29)(11,32,12,31)(13,33,14,34)(15,36,16,35)(17,39,18,40)(19,37,20,38)$
4A-1 $4^{10}$ $1$ $4$ $30$ $( 1,24, 2,23)( 3,21, 4,22)( 5,26, 6,25)( 7,27, 8,28)( 9,29,10,30)(11,31,12,32)(13,34,14,33)(15,35,16,36)(17,40,18,39)(19,38,20,37)$
4B1 $4^{10}$ $5$ $4$ $30$ $( 1,22, 2,21)( 3,23, 4,24)( 5,19, 6,20)( 7,18, 8,17)( 9,16,10,15)(11,14,12,13)(25,37,26,38)(27,39,28,40)(29,36,30,35)(31,33,32,34)$
4B-1 $4^{10}$ $5$ $4$ $30$ $( 1,14, 2,13)( 3,15, 4,16)( 5,11, 6,12)( 7,10, 8, 9)(17,37,18,38)(19,39,20,40)(21,35,22,36)(23,34,24,33)(25,32,26,31)(27,30,28,29)$
5A1 $5^{8}$ $2$ $5$ $32$ $( 1,28,12,36,19)( 2,27,11,35,20)( 3,25, 9,34,18)( 4,26,10,33,17)( 5,29,14,39,21)( 6,30,13,40,22)( 7,32,15,38,24)( 8,31,16,37,23)$
5A2 $5^{8}$ $2$ $5$ $32$ $( 1,12,19,28,36)( 2,11,20,27,35)( 3, 9,18,25,34)( 4,10,17,26,33)( 5,14,21,29,39)( 6,13,22,30,40)( 7,15,24,32,38)( 8,16,23,31,37)$
10A1 $10^{4}$ $2$ $10$ $36$ $( 1,35,28,20,12, 2,36,27,19,11)( 3,33,25,17, 9, 4,34,26,18,10)( 5,40,29,22,14, 6,39,30,21,13)( 7,37,32,23,15, 8,38,31,24,16)$
10A3 $10^{4}$ $2$ $10$ $36$ $( 1,20,36,11,28, 2,19,35,12,27)( 3,17,34,10,25, 4,18,33, 9,26)( 5,22,39,13,29, 6,21,40,14,30)( 7,23,38,16,32, 8,24,37,15,31)$
20A1 $20^{2}$ $2$ $20$ $38$ $( 1,37,35,32,28,23,20,15,12, 8, 2,38,36,31,27,24,19,16,11, 7)( 3,40,33,29,25,22,17,14, 9, 6, 4,39,34,30,26,21,18,13,10, 5)$
20A-1 $20^{2}$ $2$ $20$ $38$ $( 1,38,35,31,28,24,20,16,12, 7, 2,37,36,32,27,23,19,15,11, 8)( 3,39,33,30,25,21,17,13, 9, 5, 4,40,34,29,26,22,18,14,10, 6)$
20A3 $20^{2}$ $2$ $20$ $38$ $( 1,32,20, 8,36,24,11,37,28,15, 2,31,19, 7,35,23,12,38,27,16)( 3,29,17, 6,34,21,10,40,25,14, 4,30,18, 5,33,22, 9,39,26,13)$
20A-3 $20^{2}$ $2$ $20$ $38$ $( 1,31,20, 7,36,23,11,38,28,16, 2,32,19, 8,35,24,12,37,27,15)( 3,30,17, 5,34,22,10,39,25,13, 4,29,18, 6,33,21, 9,40,26,14)$

Malle's constant $a(G)$:     $1/20$

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 4A1 4A-1 4B1 4B-1 5A1 5A2 10A1 10A3 20A1 20A-1 20A3 20A-3
Size 1 1 5 5 1 1 5 5 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 2A 2A 2A 5A2 5A1 5A1 5A2 10A1 10A1 10A3 10A3
5 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 1A 1A 2A 2A 4A1 4A-1 4A-1 4A1
Type
40.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
40.5.1b R 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
40.5.1c R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1
40.5.1d R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
40.5.1e1 C 1 −1 −1 1 −i i i −i 1 1 −1 −1 −i i i −i
40.5.1e2 C 1 −1 −1 1 i −i −i i 1 1 −1 −1 i −i −i i
40.5.1f1 C 1 −1 1 −1 −i i −i i 1 1 −1 −1 −i i i −i
40.5.1f2 C 1 −1 1 −1 i −i i −i 1 1 −1 −1 i −i −i i
40.5.2a1 R 2 2 0 0 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
40.5.2a2 R 2 2 0 0 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
40.5.2b1 R 2 2 0 0 −2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
40.5.2b2 R 2 2 0 0 −2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
40.5.2c1 C 2 −2 0 0 −2ζ205 2ζ205 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ20−4−ζ204 ζ20−2+ζ202 ζ203+ζ207 −ζ203−ζ207 ζ203−ζ205+ζ207 −ζ203+ζ205−ζ207
40.5.2c2 C 2 −2 0 0 2ζ205 −2ζ205 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ20−4−ζ204 ζ20−2+ζ202 −ζ203−ζ207 ζ203+ζ207 −ζ203+ζ205−ζ207 ζ203−ζ205+ζ207
40.5.2c3 C 2 −2 0 0 −2ζ205 2ζ205 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 ζ20−2+ζ202 −ζ20−4−ζ204 −ζ203+ζ205−ζ207 ζ203−ζ205+ζ207 −ζ203−ζ207 ζ203+ζ207
40.5.2c4 C 2 −2 0 0 2ζ205 −2ζ205 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 ζ20−2+ζ202 −ζ20−4−ζ204 ζ203−ζ205+ζ207 −ζ203+ζ205−ζ207 ζ203+ζ207 −ζ203−ζ207

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