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Properties

Label 41T7
41T7 1 2 1->2 36 1->36 3 2->3 31 2->31 4 3->4 26 3->26 5 4->5 21 4->21 6 5->6 16 5->16 7 6->7 11 6->11 7->6 8 7->8 8->1 9 8->9 10 9->10 37 9->37 10->11 32 10->32 12 11->12 27 11->27 13 12->13 22 12->22 14 13->14 17 13->17 14->12 15 14->15 15->7 15->16 16->2 16->17 18 17->18 38 17->38 19 18->19 33 18->33 20 19->20 28 19->28 20->21 23 20->23 21->18 21->22 22->13 22->23 23->8 24 23->24 24->3 25 24->25 25->26 39 25->39 26->27 34 26->34 27->28 29 27->29 28->24 28->29 29->19 30 29->30 30->14 30->31 31->9 31->32 32->4 32->33 33->34 40 33->40 35 34->35 34->35 35->30 35->36 36->25 36->37 37->20 37->38 38->15 38->39 39->10 39->40 40->5 41 40->41 41->1
Degree $41$
Order $820$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive yes
$p$-group no
Group: $C_{41}:C_{20}$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(41, 7);
 
Copy content sage:G = TransitiveGroup(41, 7)
 
Copy content oscar:G = transitive_group(41, 7)
 
Copy content gap:G := TransitiveGroup(41, 7);
 

Group invariants

Abstract group:  $C_{41}:C_{20}$
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:  $820=2^{2} \cdot 5 \cdot 41$
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 $n$:  $41$
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 $t$:  $7$
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(41).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(41), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(41), G));
 
Generators:  $(1,36,25,39,10,32,4,21,18,33,40,5,16,2,31,9,37,20,23,8)(3,26,34,35,30,14,12,22,13,17,38,15,7,6,11,27,29,19,28,24)$, $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$
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)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$10$:  $C_{10}$
$20$:  20T1

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{41}$ $1$ $1$ $0$ $()$
2A $2^{20},1$ $41$ $2$ $20$ $( 1, 7)( 2, 6)( 3, 5)( 8,41)( 9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)$
4A1 $4^{10},1$ $41$ $4$ $30$ $( 1,18, 7,31)( 2,27, 6,22)( 3,36, 5,13)( 8,40,41, 9)(10,17,39,32)(11,26,38,23)(12,35,37,14)(15,21,34,28)(16,30,33,19)(20,25,29,24)$
4A-1 $4^{10},1$ $41$ $4$ $30$ $( 1,31, 7,18)( 2,22, 6,27)( 3,13, 5,36)( 8, 9,41,40)(10,32,39,17)(11,23,38,26)(12,14,37,35)(15,28,34,21)(16,19,33,30)(20,24,29,25)$
5A1 $5^{8},1$ $41$ $5$ $32$ $( 1,38,15,16,32)( 2,13,25,12, 9)( 3,29,35, 8,27)( 5,20,14,41,22)( 6,36,24,37,40)( 7,11,34,33,17)(10,18,23,21,30)(19,39,31,26,28)$
5A-1 $5^{8},1$ $41$ $5$ $32$ $( 1,32,16,15,38)( 2, 9,12,25,13)( 3,27, 8,35,29)( 5,22,41,14,20)( 6,40,37,24,36)( 7,17,33,34,11)(10,30,21,23,18)(19,28,26,31,39)$
5A2 $5^{8},1$ $41$ $5$ $32$ $( 1,15,32,38,16)( 2,25, 9,13,12)( 3,35,27,29, 8)( 5,14,22,20,41)( 6,24,40,36,37)( 7,34,17,11,33)(10,23,30,18,21)(19,31,28,39,26)$
5A-2 $5^{8},1$ $41$ $5$ $32$ $( 1,16,38,32,15)( 2,12,13, 9,25)( 3, 8,29,27,35)( 5,41,20,22,14)( 6,37,36,40,24)( 7,33,11,17,34)(10,21,18,30,23)(19,26,39,28,31)$
10A1 $10^{4},1$ $41$ $10$ $36$ $( 1,33,38,17,15, 7,16,11,32,34)( 2,37,13,40,25, 6,12,36, 9,24)( 3,41,29,22,35, 5, 8,20,27,14)(10,28,18,19,23,39,21,31,30,26)$
10A-1 $10^{4},1$ $41$ $10$ $36$ $( 1,34,32,11,16, 7,15,17,38,33)( 2,24, 9,36,12, 6,25,40,13,37)( 3,14,27,20, 8, 5,35,22,29,41)(10,26,30,31,21,39,23,19,18,28)$
10A3 $10^{4},1$ $41$ $10$ $36$ $( 1,17,16,34,38, 7,32,33,15,11)( 2,40,12,24,13, 6, 9,37,25,36)( 3,22, 8,14,29, 5,27,41,35,20)(10,19,21,26,18,39,30,28,23,31)$
10A-3 $10^{4},1$ $41$ $10$ $36$ $( 1,11,15,33,32, 7,38,34,16,17)( 2,36,25,37, 9, 6,13,24,12,40)( 3,20,35,41,27, 5,29,14, 8,22)(10,31,23,28,30,39,18,26,21,19)$
20A1 $20^{2},1$ $41$ $20$ $38$ $( 1,10,33,28,38,18,17,19,15,23, 7,39,16,21,11,31,32,30,34,26)( 2, 8,37,20,13,27,40,14,25, 3, 6,41,12,29,36,22, 9,35,24, 5)$
20A-1 $20^{2},1$ $41$ $20$ $38$ $( 1,26,34,30,32,31,11,21,16,39, 7,23,15,19,17,18,38,28,33,10)( 2, 5,24,35, 9,22,36,29,12,41, 6, 3,25,14,40,27,13,20,37, 8)$
20A3 $20^{2},1$ $41$ $20$ $38$ $( 1,28,17,23,16,31,34,10,38,19, 7,21,32,26,33,18,15,39,11,30)( 2,20,40, 3,12,22,24, 8,13,14, 6,29, 9, 5,37,27,25,41,36,35)$
20A-3 $20^{2},1$ $41$ $20$ $38$ $( 1,30,11,39,15,18,33,26,32,21, 7,19,38,10,34,31,16,23,17,28)( 2,35,36,41,25,27,37, 5, 9,29, 6,14,13, 8,24,22,12, 3,40,20)$
20A7 $20^{2},1$ $41$ $20$ $38$ $( 1,19,11,10,15,31,33,23,32,28, 7,30,38,39,34,18,16,26,17,21)( 2,14,36, 8,25,22,37, 3, 9,20, 6,35,13,41,24,27,12, 5,40,29)$
20A-7 $20^{2},1$ $41$ $20$ $38$ $( 1,21,17,26,16,18,34,39,38,30, 7,28,32,23,33,31,15,10,11,19)( 2,29,40, 5,12,27,24,41,13,35, 6,20, 9, 3,37,22,25, 8,36,14)$
20A9 $20^{2},1$ $41$ $20$ $38$ $( 1,23,34,19,32,18,11,28,16,10, 7,26,15,30,17,31,38,21,33,39)( 2, 3,24,14, 9,27,36,20,12, 8, 6, 5,25,35,40,22,13,29,37,41)$
20A-9 $20^{2},1$ $41$ $20$ $38$ $( 1,39,33,21,38,31,17,30,15,26, 7,10,16,28,11,18,32,19,34,23)( 2,41,37,29,13,22,40,35,25, 5, 6, 8,12,20,36,27, 9,14,24, 3)$
41A1 $41$ $20$ $41$ $40$ $( 1, 6,11,16,21,26,31,36,41, 5,10,15,20,25,30,35,40, 4, 9,14,19,24,29,34,39, 3, 8,13,18,23,28,33,38, 2, 7,12,17,22,27,32,37)$
41A3 $41$ $20$ $41$ $40$ $( 1,16,31, 5,20,35, 9,24,39,13,28, 2,17,32, 6,21,36,10,25,40,14,29, 3,18,33, 7,22,37,11,26,41,15,30, 4,19,34, 8,23,38,12,27)$

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 4A1 4A-1 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 20A1 20A-1 20A3 20A-3 20A7 20A-7 20A9 20A-9 41A1 41A3
Size 1 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 20 20
2 P 1A 1A 2A 2A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 10A1 10A-1 10A3 10A-3 10A-3 10A3 10A-1 10A1 41A1 41A3
5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 2A 2A 2A 2A 4A1 4A-1 4A-1 4A1 4A-1 4A1 4A1 4A-1 41A1 41A3
41 P 1A 2A 4A1 4A-1 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 20A1 20A-1 20A3 20A-3 20A7 20A-7 20A9 20A-9 1A 1A
Type
820.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
820.7.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1
820.7.1c1 C 1 −1 −i i 1 1 1 1 −1 −1 −1 −1 i −i i −i −i i −i i 1 1
820.7.1c2 C 1 −1 i −i 1 1 1 1 −1 −1 −1 −1 −i i −i i i −i i −i 1 1
820.7.1d1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ52 ζ52 ζ5 ζ5 ζ5−1 ζ5−1 ζ5−2 ζ5−2 1 1
820.7.1d2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−2 ζ5−2 ζ5−1 ζ5−1 ζ5 ζ5 ζ52 ζ52 1 1
820.7.1d3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5 ζ5 ζ5−2 ζ5−2 ζ52 ζ52 ζ5−1 ζ5−1 1 1
820.7.1d4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−1 ζ5−1 ζ52 ζ52 ζ5−2 ζ5−2 ζ5 ζ5 1 1
820.7.1e1 C 1 1 −1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ52 −ζ52 −ζ5 −ζ5 −ζ5−1 −ζ5−1 −ζ5−2 −ζ5−2 1 1
820.7.1e2 C 1 1 −1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5−2 −ζ5−2 −ζ5−1 −ζ5−1 −ζ5 −ζ5 −ζ52 −ζ52 1 1
820.7.1e3 C 1 1 −1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5 −ζ5 −ζ5−2 −ζ5−2 −ζ52 −ζ52 −ζ5−1 −ζ5−1 1 1
820.7.1e4 C 1 1 −1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5−1 −ζ5−1 −ζ52 −ζ52 −ζ5−2 −ζ5−2 −ζ5 −ζ5 1 1
820.7.1f1 C 1 −1 −ζ205 ζ205 −ζ202 ζ208 ζ204 −ζ206 ζ206 −ζ204 −ζ208 ζ202 −ζ203 ζ203 ζ209 −ζ209 −ζ20 ζ20 ζ207 −ζ207 1 1
820.7.1f2 C 1 −1 ζ205 −ζ205 ζ208 −ζ202 −ζ206 ζ204 −ζ204 ζ206 ζ202 −ζ208 ζ207 −ζ207 −ζ20 ζ20 ζ209 −ζ209 −ζ203 ζ203 1 1
820.7.1f3 C 1 −1 −ζ205 ζ205 ζ208 −ζ202 −ζ206 ζ204 −ζ204 ζ206 ζ202 −ζ208 −ζ207 ζ207 ζ20 −ζ20 −ζ209 ζ209 ζ203 −ζ203 1 1
820.7.1f4 C 1 −1 ζ205 −ζ205 −ζ202 ζ208 ζ204 −ζ206 ζ206 −ζ204 −ζ208 ζ202 ζ203 −ζ203 −ζ209 ζ209 ζ20 −ζ20 −ζ207 ζ207 1 1
820.7.1f5 C 1 −1 −ζ205 ζ205 −ζ206 ζ204 −ζ202 ζ208 −ζ208 ζ202 −ζ204 ζ206 ζ209 −ζ209 −ζ207 ζ207 ζ203 −ζ203 −ζ20 ζ20 1 1
820.7.1f6 C 1 −1 ζ205 −ζ205 ζ204 −ζ206 ζ208 −ζ202 ζ202 −ζ208 ζ206 −ζ204 −ζ20 ζ20 ζ203 −ζ203 −ζ207 ζ207 ζ209 −ζ209 1 1
820.7.1f7 C 1 −1 −ζ205 ζ205 ζ204 −ζ206 ζ208 −ζ202 ζ202 −ζ208 ζ206 −ζ204 ζ20 −ζ20 −ζ203 ζ203 ζ207 −ζ207 −ζ209 ζ209 1 1
820.7.1f8 C 1 −1 ζ205 −ζ205 −ζ206 ζ204 −ζ202 ζ208 −ζ208 ζ202 −ζ204 ζ206 −ζ209 ζ209 ζ207 −ζ207 −ζ203 ζ203 ζ20 −ζ20 1 1
820.7.20a1 R 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ζ41−19+ζ41−17+ζ41−15+ζ41−14+ζ41−13+ζ41−12+ζ41−11+ζ41−7+ζ41−6+ζ41−3+ζ413+ζ416+ζ417+ζ4111+ζ4112+ζ4113+ζ4114+ζ4115+ζ4117+ζ4119 −ζ41−19−ζ41−17−ζ41−15−ζ41−14−ζ41−13−ζ41−12−ζ41−11−ζ41−7−ζ41−6−ζ41−3−1−ζ413−ζ416−ζ417−ζ4111−ζ4112−ζ4113−ζ4114−ζ4115−ζ4117−ζ4119
820.7.20a2 R 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −ζ41−19−ζ41−17−ζ41−15−ζ41−14−ζ41−13−ζ41−12−ζ41−11−ζ41−7−ζ41−6−ζ41−3−1−ζ413−ζ416−ζ417−ζ4111−ζ4112−ζ4113−ζ4114−ζ4115−ζ4117−ζ4119 ζ41−19+ζ41−17+ζ41−15+ζ41−14+ζ41−13+ζ41−12+ζ41−11+ζ41−7+ζ41−6+ζ41−3+ζ413+ζ416+ζ417+ζ4111+ζ4112+ζ4113+ζ4114+ζ4115+ζ4117+ζ4119

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