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Properties

Label 40T464
40T464 1 15 1->15 1->15 2 16 2->16 2->16 3 13 3->13 3->13 4 14 4->14 4->14 5 12 5->12 20 5->20 6 11 6->11 19 6->19 7 9 7->9 17 7->17 8 10 8->10 18 8->18 9->6 22 9->22 10->5 21 10->21 11->7 23 11->23 12->8 24 12->24 13->1 25 13->25 14->2 26 14->26 15->3 27 15->27 16->4 28 16->28 30 17->30 37 17->37 29 18->29 38 18->38 31 19->31 40 19->40 32 20->32 39 20->39 36 21->36 21->36 35 22->35 22->35 34 23->34 23->34 33 24->33 24->33 25->32 25->40 26->31 26->39 27->29 27->37 28->30 28->38 29->1 29->25 30->2 30->26 31->4 31->28 32->3 32->27 33->5 33->21 34->6 34->22 35->7 35->23 36->8 36->24 37->12 37->19 38->11 38->20 39->9 39->18 40->10 40->17
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, 464);
 
Copy content sage:G = TransitiveGroup(40, 464)
 
Copy content oscar:G = transitive_group(40, 464)
 
Copy content gap:G := TransitiveGroup(40, 464);
 

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 $t$:  $464$
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:  $(1,15,3,13)(2,16,4,14)(5,12,8,10)(6,11,7,9)(17,37,19,40)(18,38,20,39)(21,36,24,33)(22,35,23,34)(25,32,27,29)(26,31,28,30)$, $(1,15,27,37,12,24,33,5,20,32,3,13,25,40,10,21,36,8,18,29)(2,16,28,38,11,23,34,6,19,31,4,14,26,39,9,22,35,7,17,30)$
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: $C_2$

Degree 4: $C_4$

Degree 5: $D_{5}$

Degree 8: None

Degree 10: $D_{10}$, $(C_2^4 : C_5) : C_2$, $C_2\times (C_2^4 : D_5)$

Degree 20: 20T6, 20T71, 20T144

Low degree siblings

20T144 x 6, 40T455 x 3, 40T464 x 5, 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 $2^{20}$ $1$ $2$ $20$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)$
2B $2^{16},1^{8}$ $5$ $2$ $16$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
2C $2^{8},1^{24}$ $5$ $2$ $8$ $( 5, 6)( 7, 8)( 9,10)(11,12)(25,26)(27,28)(29,30)(31,32)$
2D $2^{8},1^{24}$ $5$ $2$ $8$ $( 1, 2)( 3, 4)(13,14)(15,16)(21,22)(23,24)(33,34)(35,36)$
2E $2^{20}$ $5$ $2$ $20$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)$
2F $2^{20}$ $5$ $2$ $20$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)(37,40)(38,39)$
2G $2^{20}$ $5$ $2$ $20$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)(37,40)(38,39)$
2H $2^{20}$ $20$ $2$ $20$ $( 1, 2)( 3, 4)( 5,37)( 6,38)( 7,39)( 8,40)( 9,34)(10,33)(11,35)(12,36)(13,29)(14,30)(15,32)(16,31)(17,28)(18,27)(19,26)(20,25)(21,22)(23,24)$
2I $2^{20}$ $20$ $2$ $20$ $( 1, 4)( 2, 3)( 5,40)( 6,39)( 7,38)( 8,37)( 9,35)(10,36)(11,34)(12,33)(13,32)(14,31)(15,29)(16,30)(17,26)(18,25)(19,28)(20,27)(21,23)(22,24)$
4A1 $4^{10}$ $1$ $4$ $30$ $( 1,24, 3,21)( 2,23, 4,22)( 5,25, 8,27)( 6,26, 7,28)( 9,30,11,31)(10,29,12,32)(13,36,15,33)(14,35,16,34)(17,38,19,39)(18,37,20,40)$
4A-1 $4^{10}$ $1$ $4$ $30$ $( 1,21, 3,24)( 2,22, 4,23)( 5,27, 8,25)( 6,28, 7,26)( 9,31,11,30)(10,32,12,29)(13,33,15,36)(14,34,16,35)(17,39,19,38)(18,40,20,37)$
4B1 $4^{10}$ $5$ $4$ $30$ $( 1,24, 3,21)( 2,23, 4,22)( 5,26, 8,28)( 6,25, 7,27)( 9,30,11,31)(10,29,12,32)(13,36,15,33)(14,35,16,34)(17,37,19,40)(18,38,20,39)$
4B-1 $4^{10}$ $5$ $4$ $30$ $( 1,21, 3,24)( 2,22, 4,23)( 5,28, 8,26)( 6,27, 7,25)( 9,31,11,30)(10,32,12,29)(13,33,15,36)(14,34,16,35)(17,40,19,37)(18,39,20,38)$
4C1 $4^{10}$ $5$ $4$ $30$ $( 1,24, 3,21)( 2,23, 4,22)( 5,26, 8,28)( 6,25, 7,27)( 9,29,11,32)(10,30,12,31)(13,36,15,33)(14,35,16,34)(17,38,19,39)(18,37,20,40)$
4C-1 $4^{10}$ $5$ $4$ $30$ $( 1,21, 3,24)( 2,22, 4,23)( 5,28, 8,26)( 6,27, 7,25)( 9,32,11,29)(10,31,12,30)(13,33,15,36)(14,34,16,35)(17,39,19,38)(18,40,20,37)$
4D1 $4^{10}$ $5$ $4$ $30$ $( 1,24, 3,21)( 2,23, 4,22)( 5,26, 8,28)( 6,25, 7,27)( 9,29,11,32)(10,30,12,31)(13,35,15,34)(14,36,16,33)(17,37,19,40)(18,38,20,39)$
4D-1 $4^{10}$ $5$ $4$ $30$ $( 1,21, 3,24)( 2,22, 4,23)( 5,28, 8,26)( 6,27, 7,25)( 9,32,11,29)(10,31,12,30)(13,34,15,35)(14,33,16,36)(17,40,19,37)(18,39,20,38)$
4E $4^{8},2^{4}$ $20$ $4$ $28$ $( 1, 2)( 3, 4)( 5,37, 6,38)( 7,39, 8,40)( 9,33,10,34)(11,36,12,35)(13,29,14,30)(15,32,16,31)(17,27,18,28)(19,25,20,26)(21,22)(23,24)$
4F $4^{8},2^{4}$ $20$ $4$ $28$ $( 1, 4)( 2, 3)( 5,40, 6,39)( 7,38, 8,37)( 9,36,10,35)(11,33,12,34)(13,32,14,31)(15,29,16,30)(17,25,18,26)(19,27,20,28)(21,23)(22,24)$
4G $4^{4},2^{8},1^{8}$ $20$ $4$ $20$ $( 1,35)( 2,36)( 3,34)( 4,33)( 5,30, 6,29)( 7,32, 8,31)( 9,28,10,27)(11,26,12,25)(13,22)(14,21)(15,23)(16,24)$
4H $4^{4},2^{12}$ $20$ $4$ $24$ $( 1,34)( 2,33)( 3,35)( 4,36)( 5,31, 6,32)( 7,29, 8,30)( 9,26,10,25)(11,28,12,27)(13,23)(14,24)(15,22)(16,21)(17,19)(18,20)(37,40)(38,39)$
4I $4^{4},2^{8},1^{8}$ $20$ $4$ $20$ $( 1,35, 2,36)( 3,34, 4,33)( 5,29)( 6,30)( 7,31)( 8,32)( 9,28)(10,27)(11,26)(12,25)(13,21,14,22)(15,24,16,23)$
4J $4^{4},2^{12}$ $20$ $4$ $24$ $( 1,34, 2,33)( 3,35, 4,36)( 5,32)( 6,31)( 7,30)( 8,29)( 9,26)(10,25)(11,28)(12,27)(13,24,14,23)(15,21,16,22)(17,19)(18,20)(37,40)(38,39)$
4K1 $4^{10}$ $20$ $4$ $30$ $( 1,23, 3,22)( 2,24, 4,21)( 5,20, 8,18)( 6,19, 7,17)( 9,14,11,16)(10,13,12,15)(25,40,27,37)(26,39,28,38)(29,36,32,33)(30,35,31,34)$
4K-1 $4^{10}$ $20$ $4$ $30$ $( 1,22, 3,23)( 2,21, 4,24)( 5,18, 8,20)( 6,17, 7,19)( 9,16,11,14)(10,15,12,13)(25,37,27,40)(26,38,28,39)(29,33,32,36)(30,34,31,35)$
4L1 $4^{10}$ $20$ $4$ $30$ $( 1,23, 3,22)( 2,24, 4,21)( 5,20, 7,17)( 6,19, 8,18)( 9,13,12,16)(10,14,11,15)(25,40,28,38)(26,39,27,37)(29,35,31,33)(30,36,32,34)$
4L-1 $4^{10}$ $20$ $4$ $30$ $( 1,22, 3,23)( 2,21, 4,24)( 5,18, 7,19)( 6,17, 8,20)( 9,15,12,14)(10,16,11,13)(25,37,28,39)(26,38,27,40)(29,34,31,36)(30,33,32,35)$
4M1 $4^{10}$ $20$ $4$ $30$ $( 1,16, 3,14)( 2,15, 4,13)( 5,11, 7,10)( 6,12, 8, 9)(17,38,19,39)(18,37,20,40)(21,35,24,34)(22,36,23,33)(25,31,28,29)(26,32,27,30)$
4M-1 $4^{10}$ $20$ $4$ $30$ $( 1,14, 3,16)( 2,13, 4,15)( 5, 9, 7,12)( 6,10, 8,11)(17,39,19,38)(18,40,20,37)(21,34,24,35)(22,33,23,36)(25,30,28,32)(26,29,27,31)$
4N1 $4^{10}$ $20$ $4$ $30$ $( 1,16, 4,13)( 2,15, 3,14)( 5,12, 8,10)( 6,11, 7, 9)(17,38,19,39)(18,37,20,40)(21,35,23,33)(22,36,24,34)(25,32,27,29)(26,31,28,30)$
4N-1 $4^{10}$ $20$ $4$ $30$ $( 1,14, 4,15)( 2,13, 3,16)( 5,10, 8,12)( 6, 9, 7,11)(17,39,19,38)(18,40,20,37)(21,34,23,36)(22,33,24,35)(25,29,27,32)(26,30,28,31)$
5A1 $5^{8}$ $32$ $5$ $32$ $( 1,11,19,25,35)( 2,12,20,26,36)( 3, 9,17,27,34)( 4,10,18,28,33)( 5,14,21,30,38)( 6,13,22,29,37)( 7,15,23,32,40)( 8,16,24,31,39)$
5A2 $5^{8}$ $32$ $5$ $32$ $( 1,19,35,11,25)( 2,20,36,12,26)( 3,17,34, 9,27)( 4,18,33,10,28)( 5,21,38,14,30)( 6,22,37,13,29)( 7,23,40,15,32)( 8,24,39,16,31)$
10A1 $10^{4}$ $32$ $10$ $36$ $( 1,27,11,34,19, 3,25, 9,35,17)( 2,28,12,33,20, 4,26,10,36,18)( 5,31,14,39,21, 8,30,16,38,24)( 6,32,13,40,22, 7,29,15,37,23)$
10A3 $10^{4}$ $32$ $10$ $36$ $( 1,34,25,17,11, 3,35,27,19, 9)( 2,33,26,18,12, 4,36,28,20,10)( 5,39,30,24,14, 8,38,31,21,16)( 6,40,29,23,13, 7,37,32,22,15)$
20A1 $20^{2}$ $32$ $20$ $38$ $( 1,16,27,38,11,24,34, 5,19,31, 3,14,25,39, 9,21,35, 8,17,30)( 2,15,28,37,12,23,33, 6,20,32, 4,13,26,40,10,22,36, 7,18,29)$
20A-1 $20^{2}$ $32$ $20$ $38$ $( 1,30,17, 8,35,21, 9,39,25,14, 3,31,19, 5,34,24,11,38,27,16)( 2,29,18, 7,36,22,10,40,26,13, 4,32,20, 6,33,23,12,37,28,15)$
20A3 $20^{2}$ $32$ $20$ $38$ $( 1,38,34,31,25,21,17,16,11, 5, 3,39,35,30,27,24,19,14, 9, 8)( 2,37,33,32,26,22,18,15,12, 6, 4,40,36,29,28,23,20,13,10, 7)$
20A-3 $20^{2}$ $32$ $20$ $38$ $( 1, 8, 9,14,19,24,27,30,35,39, 3, 5,11,16,17,21,25,31,34,38)( 2, 7,10,13,20,23,28,29,36,40, 4, 6,12,15,18,22,26,32,33,37)$

Malle's constant $a(G)$:     $1/8$

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

Data not computed