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Properties

Label 40T455
40T455 1 32 1->32 34 1->34 2 31 2->31 33 2->33 3 29 3->29 36 3->36 4 30 4->30 35 4->35 5 5->32 5->33 6 6->31 6->34 7 7->29 7->35 8 8->30 8->36 9 27 9->27 40 9->40 10 28 10->28 39 10->39 11 26 11->26 37 11->37 12 25 12->25 38 12->38 13 13->3 23 13->23 14 14->4 24 14->24 15 15->1 22 15->22 16 16->2 21 16->21 17 17->6 19 17->19 18 18->5 20 18->20 19->7 19->18 20->8 20->17 21->9 22->10 23->12 23->14 24->11 24->13 25->13 26->14 27->15 28->16 29->18 30->17 31->20 32->19 33->1 33->22 34->2 34->21 35->24 36->23 37->28 37->39 38->27 38->40 39->26 39->38 40->25 40->37
Degree $40$
Order $640$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2^5.D_{10}$

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, 455);
 
Copy content sage:G = TransitiveGroup(40, 455)
 
Copy content oscar:G = transitive_group(40, 455)
 
Copy content gap:G := TransitiveGroup(40, 455);
 

Group invariants

Abstract group:  $C_2^5.D_{10}$
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:  $640=2^{7} \cdot 5$
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 $t$:  $455$
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:  $(1,34,2,33)(3,36)(4,35)(5,32)(6,31)(7,29)(8,30)(9,27)(10,28)(11,26)(12,25)(13,23,14,24)(15,22)(16,21)(17,19,18,20)(37,39,38,40)$, $(1,32,19,7,35,24,11,37,28,16,2,31,20,8,36,23,12,38,27,15)(3,29,18,5,33,22,10,39,26,14,4,30,17,6,34,21,9,40,25,13)$
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, $C_2^2$
$8$:  $C_4\times C_2$
$10$:  $D_{5}$
$20$:  $D_{10}$
$40$:  20T6
$160$:  $(C_2^4 : C_5) : C_2$
$320$:  $C_2\times (C_2^4 : D_5)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 8: None

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

Degree 20: 20T4, 20T144 x 2

Low degree siblings

20T144 x 6, 40T455 x 2, 40T464 x 6, 40T465 x 6, 40T533 x 6, 40T535 x 6, 40T544 x 6, 40T545 x 6

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 $2^{8},1^{24}$ $5$ $2$ $8$ $( 3, 4)( 7, 8)( 9,10)(15,16)(21,22)(27,28)(29,30)(35,36)$
2C $2^{16},1^{8}$ $5$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(31,32)(33,34)(37,38)(39,40)$
2D $2^{8},1^{24}$ $5$ $2$ $8$ $( 1, 2)( 3, 4)( 7, 8)(17,18)(21,22)(23,24)(27,28)(39,40)$
2E $2^{12},1^{16}$ $5$ $2$ $12$ $( 1, 2)( 5, 6)(11,12)(13,14)(17,18)(19,20)(23,24)(25,26)(31,32)(33,34)(37,38)(39,40)$
2F $2^{4},1^{32}$ $5$ $2$ $4$ $( 9,10)(15,16)(29,30)(35,36)$
2G $2^{12},1^{16}$ $5$ $2$ $12$ $( 5, 6)( 9,10)(11,12)(13,14)(15,16)(19,20)(25,26)(29,30)(31,32)(33,34)(35,36)(37,38)$
2H $2^{20}$ $20$ $2$ $20$ $( 1,33)( 2,34)( 3,36)( 4,35)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,25)(12,26)(13,23)(14,24)(15,22)(16,21)(17,20)(18,19)(37,39)(38,40)$
2I $2^{20}$ $20$ $2$ $20$ $( 1,34)( 2,33)( 3,35)( 4,36)( 5,32)( 6,31)( 7,30)( 8,29)( 9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,21)(16,22)(17,19)(18,20)(37,40)(38,39)$
4A1 $4^{10}$ $1$ $4$ $30$ $( 1,24, 2,23)( 3,22, 4,21)( 5,26, 6,25)( 7,28, 8,27)( 9,29,10,30)(11,31,12,32)(13,33,14,34)(15,35,16,36)(17,40,18,39)(19,37,20,38)$
4A-1 $4^{10}$ $1$ $4$ $30$ $( 1,23, 2,24)( 3,21, 4,22)( 5,25, 6,26)( 7,27, 8,28)( 9,30,10,29)(11,32,12,31)(13,34,14,33)(15,36,16,35)(17,39,18,40)(19,38,20,37)$
4B1 $4^{10}$ $5$ $4$ $30$ $( 1,23, 2,24)( 3,22, 4,21)( 5,26, 6,25)( 7,28, 8,27)( 9,29,10,30)(11,31,12,32)(13,34,14,33)(15,35,16,36)(17,39,18,40)(19,38,20,37)$
4B-1 $4^{10}$ $5$ $4$ $30$ $( 1,24, 2,23)( 3,21, 4,22)( 5,25, 6,26)( 7,27, 8,28)( 9,30,10,29)(11,32,12,31)(13,33,14,34)(15,36,16,35)(17,40,18,39)(19,37,20,38)$
4C1 $4^{10}$ $5$ $4$ $30$ $( 1,23, 2,24)( 3,21, 4,22)( 5,25, 6,26)( 7,27, 8,28)( 9,29,10,30)(11,32,12,31)(13,34,14,33)(15,35,16,36)(17,39,18,40)(19,38,20,37)$
4C-1 $4^{10}$ $5$ $4$ $30$ $( 1,24, 2,23)( 3,22, 4,21)( 5,26, 6,25)( 7,28, 8,27)( 9,30,10,29)(11,31,12,32)(13,33,14,34)(15,36,16,35)(17,40,18,39)(19,37,20,38)$
4D1 $4^{10}$ $5$ $4$ $30$ $( 1,24, 2,23)( 3,21, 4,22)( 5,26, 6,25)( 7,27, 8,28)( 9,30,10,29)(11,31,12,32)(13,33,14,34)(15,36,16,35)(17,40,18,39)(19,37,20,38)$
4D-1 $4^{10}$ $5$ $4$ $30$ $( 1,23, 2,24)( 3,22, 4,21)( 5,25, 6,26)( 7,28, 8,27)( 9,29,10,30)(11,32,12,31)(13,34,14,33)(15,35,16,36)(17,39,18,40)(19,38,20,37)$
4E $4^{4},2^{12}$ $20$ $4$ $24$ $( 1,33)( 2,34)( 3,35, 4,36)( 5,32)( 6,31)( 7,30, 8,29)( 9,27,10,28)(11,26)(12,25)(13,23)(14,24)(15,22,16,21)(17,20)(18,19)(37,39)(38,40)$
4F $4^{4},2^{12}$ $20$ $4$ $24$ $( 1,34)( 2,33)( 3,36, 4,35)( 5,31)( 6,32)( 7,29, 8,30)( 9,28,10,27)(11,25)(12,26)(13,24)(14,23)(15,21,16,22)(17,19)(18,20)(37,40)(38,39)$
4G $4^{8},2^{4}$ $20$ $4$ $28$ $( 1, 4, 2, 3)( 5,38, 6,37)( 7,39, 8,40)( 9,35)(10,36)(11,33,12,34)(13,31,14,32)(15,30)(16,29)(17,27,18,28)(19,25,20,26)(21,23,22,24)$
4H $4^{8},2^{4}$ $20$ $4$ $28$ $( 1, 3, 2, 4)( 5,37, 6,38)( 7,40, 8,39)( 9,36)(10,35)(11,34,12,33)(13,32,14,31)(15,29)(16,30)(17,28,18,27)(19,26,20,25)(21,24,22,23)$
4I $4^{4},2^{12}$ $20$ $4$ $24$ $( 1, 4, 2, 3)( 5,37)( 6,38)( 7,39, 8,40)( 9,36)(10,35)(11,34)(12,33)(13,31)(14,32)(15,29)(16,30)(17,27,18,28)(19,25)(20,26)(21,23,22,24)$
4J $4^{4},2^{12}$ $20$ $4$ $24$ $( 1, 3, 2, 4)( 5,38)( 6,37)( 7,40, 8,39)( 9,35)(10,36)(11,33)(12,34)(13,32)(14,31)(15,30)(16,29)(17,28,18,27)(19,26)(20,25)(21,24,22,23)$
4K1 $4^{10}$ $20$ $4$ $30$ $( 1,14, 2,13)( 3,15, 4,16)( 5,12, 6,11)( 7,10, 8, 9)(17,38,18,37)(19,39,20,40)(21,36,22,35)(23,33,24,34)(25,31,26,32)(27,29,28,30)$
4K-1 $4^{10}$ $20$ $4$ $30$ $( 1,13, 2,14)( 3,16, 4,15)( 5,11, 6,12)( 7, 9, 8,10)(17,37,18,38)(19,40,20,39)(21,35,22,36)(23,34,24,33)(25,32,26,31)(27,30,28,29)$
4L1 $4^{6},2^{8}$ $20$ $4$ $26$ $( 1,14, 2,13)( 3,16)( 4,15)( 5,11, 6,12)( 7, 9)( 8,10)(17,38,18,37)(19,39,20,40)(21,35)(22,36)(23,33,24,34)(25,32,26,31)(27,30)(28,29)$
4L-1 $4^{6},2^{8}$ $20$ $4$ $26$ $( 1,13, 2,14)( 3,15)( 4,16)( 5,12, 6,11)( 7,10)( 8, 9)(17,37,18,38)(19,40,20,39)(21,36)(22,35)(23,34,24,33)(25,31,26,32)(27,29)(28,30)$
4M1 $4^{2},2^{16}$ $20$ $4$ $22$ $( 1,21)( 2,22)( 3,24)( 4,23)( 5,19)( 6,20)( 7,17)( 8,18)( 9,16,10,15)(11,14)(12,13)(25,38)(26,37)(27,39)(28,40)(29,36,30,35)(31,34)(32,33)$
4M-1 $4^{2},2^{16}$ $20$ $4$ $22$ $( 1,22)( 2,21)( 3,23)( 4,24)( 5,20)( 6,19)( 7,18)( 8,17)( 9,15,10,16)(11,13)(12,14)(25,37)(26,38)(27,40)(28,39)(29,35,30,36)(31,33)(32,34)$
4N1 $4^{6},2^{8}$ $20$ $4$ $26$ $( 1,21)( 2,22)( 3,24)( 4,23)( 5,20, 6,19)( 7,17)( 8,18)( 9,15,10,16)(11,13,12,14)(25,37,26,38)(27,39)(28,40)(29,35,30,36)(31,33,32,34)$
4N-1 $4^{6},2^{8}$ $20$ $4$ $26$ $( 1,22)( 2,21)( 3,23)( 4,24)( 5,19, 6,20)( 7,18)( 8,17)( 9,16,10,15)(11,14,12,13)(25,38,26,37)(27,40)(28,39)(29,36,30,35)(31,34,32,33)$
5A1 $5^{8}$ $32$ $5$ $32$ $( 1,36,27,19,12)( 2,35,28,20,11)( 3,33,25,18, 9)( 4,34,26,17,10)( 5,39,29,22,14)( 6,40,30,21,13)( 7,37,32,24,15)( 8,38,31,23,16)$
5A2 $5^{8}$ $32$ $5$ $32$ $( 1,27,12,36,19)( 2,28,11,35,20)( 3,25, 9,33,18)( 4,26,10,34,17)( 5,29,14,39,22)( 6,30,13,40,21)( 7,32,15,37,24)( 8,31,16,38,23)$
10A1 $10^{4}$ $32$ $10$ $36$ $( 1,20,36,11,27, 2,19,35,12,28)( 3,17,33,10,25, 4,18,34, 9,26)( 5,21,39,13,29, 6,22,40,14,30)( 7,23,37,16,32, 8,24,38,15,31)$
10A3 $10^{4}$ $32$ $10$ $36$ $( 1,11,19,28,36, 2,12,20,27,35)( 3,10,18,26,33, 4, 9,17,25,34)( 5,13,22,30,39, 6,14,21,29,40)( 7,16,24,31,37, 8,15,23,32,38)$
20A1 $20^{2}$ $32$ $20$ $38$ $( 1,32,20, 8,36,24,11,38,27,15, 2,31,19, 7,35,23,12,37,28,16)( 3,29,17, 6,33,22,10,40,25,14, 4,30,18, 5,34,21, 9,39,26,13)$
20A-1 $20^{2}$ $32$ $20$ $38$ $( 1,16,28,37,12,23,35, 7,19,31, 2,15,27,38,11,24,36, 8,20,32)( 3,13,26,39, 9,21,34, 5,18,30, 4,14,25,40,10,22,33, 6,17,29)$
20A3 $20^{2}$ $32$ $20$ $38$ $( 1, 8,11,15,19,23,28,32,36,38, 2, 7,12,16,20,24,27,31,35,37)( 3, 6,10,14,18,21,26,29,33,40, 4, 5, 9,13,17,22,25,30,34,39)$
20A-3 $20^{2}$ $32$ $20$ $38$ $( 1,37,35,31,27,24,20,16,12, 7, 2,38,36,32,28,23,19,15,11, 8)( 3,39,34,30,25,22,17,13, 9, 5, 4,40,33,29,26,21,18,14,10, 6)$

Malle's constant $a(G)$:     $1/4$

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

40 x 40 character table

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

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