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Properties

Label 40T465
40T465 1 13 1->13 28 1->28 2 14 2->14 27 2->27 3 15 3->15 26 3->26 4 16 4->16 25 4->25 5 9 5->9 23 5->23 6 10 6->10 24 6->24 7 12 7->12 21 7->21 8 11 8->11 22 8->22 9->7 19 9->19 10->8 20 10->20 11->6 17 11->17 12->5 18 12->18 13->3 13->16 14->4 14->15 15->1 16->2 17->12 39 17->39 18->11 40 18->40 19->10 38 19->38 20->9 37 20->37 33 21->33 34 22->34 35 23->35 36 24->36 30 25->30 29 26->29 31 27->31 32 28->32 29->27 29->40 30->28 30->39 31->26 31->38 32->25 32->37 33->24 33->35 34->23 34->36 35->22 36->21 37->18 37->31 38->17 38->32 39->19 39->29 40->20 40->30
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, 465);
 
Copy content sage:G = TransitiveGroup(40, 465)
 
Copy content oscar:G = transitive_group(40, 465)
 
Copy content gap:G := TransitiveGroup(40, 465);
 

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 $t$:  $465$
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:  $(1,28)(2,27)(3,26)(4,25)(5,23)(6,24)(7,21)(8,22)(9,19,10,20)(11,17,12,18)(13,16)(14,15)(29,40,30,39)(31,38,32,37)(33,35)(34,36)$, $(1,13,3,15)(2,14,4,16)(5,9,7,12)(6,10,8,11)(17,39,19,38)(18,40,20,37)(21,33,24,36)(22,34,23,35)(25,30,28,32)(26,29,27,31)$
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, 20T73, 20T144

Low degree siblings

20T144 x 6, 40T455 x 3, 40T464 x 6, 40T465 x 5, 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 $2^{8},1^{24}$ $5$ $2$ $8$ $( 5, 6)( 7, 8)( 9,10)(11,12)(25,26)(27,28)(29,30)(31,32)$
2C $2^{8},1^{24}$ $5$ $2$ $8$ $( 5, 6)( 7, 8)(17,18)(19,20)(25,26)(27,28)(37,38)(39,40)$
2D $2^{16},1^{8}$ $5$ $2$ $16$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
2E $2^{20}$ $5$ $2$ $20$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)(37,40)(38,39)$
2F $2^{20}$ $5$ $2$ $20$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)$
2G $2^{20}$ $5$ $2$ $20$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)$
2H $2^{20}$ $20$ $2$ $20$ $( 1,34)( 2,33)( 3,35)( 4,36)( 5,32)( 6,31)( 7,30)( 8,29)( 9,26)(10,25)(11,28)(12,27)(13,23)(14,24)(15,22)(16,21)(17,19)(18,20)(37,40)(38,39)$
2I $2^{16},1^{8}$ $20$ $2$ $16$ $( 1,35)( 2,36)( 3,34)( 4,33)( 5,29)( 6,30)( 7,31)( 8,32)( 9,28)(10,27)(11,26)(12,25)(13,22)(14,21)(15,23)(16,24)$
4A1 $4^{10}$ $1$ $4$ $30$ $( 1,22, 3,23)( 2,21, 4,24)( 5,28, 8,26)( 6,27, 7,25)( 9,32,11,29)(10,31,12,30)(13,34,15,35)(14,33,16,36)(17,40,19,37)(18,39,20,38)$
4A-1 $4^{10}$ $1$ $4$ $30$ $( 1,23, 3,22)( 2,24, 4,21)( 5,26, 8,28)( 6,25, 7,27)( 9,29,11,32)(10,30,12,31)(13,35,15,34)(14,36,16,33)(17,37,19,40)(18,38,20,39)$
4B1 $4^{10}$ $5$ $4$ $30$ $( 1,21, 3,24)( 2,22, 4,23)( 5,27, 8,25)( 6,28, 7,26)( 9,31,11,30)(10,32,12,29)(13,33,15,36)(14,34,16,35)(17,40,19,37)(18,39,20,38)$
4B-1 $4^{10}$ $5$ $4$ $30$ $( 1,24, 3,21)( 2,23, 4,22)( 5,25, 8,27)( 6,26, 7,28)( 9,30,11,31)(10,29,12,32)(13,36,15,33)(14,35,16,34)(17,37,19,40)(18,38,20,39)$
4C1 $4^{10}$ $5$ $4$ $30$ $( 1,22, 3,23)( 2,21, 4,24)( 5,27, 8,25)( 6,28, 7,26)( 9,32,11,29)(10,31,12,30)(13,34,15,35)(14,33,16,36)(17,39,19,38)(18,40,20,37)$
4C-1 $4^{10}$ $5$ $4$ $30$ $( 1,23, 3,22)( 2,24, 4,21)( 5,25, 8,27)( 6,26, 7,28)( 9,29,11,32)(10,30,12,31)(13,35,15,34)(14,36,16,33)(17,38,19,39)(18,37,20,40)$
4D1 $4^{10}$ $5$ $4$ $30$ $( 1,22, 3,23)( 2,21, 4,24)( 5,27, 8,25)( 6,28, 7,26)( 9,31,11,30)(10,32,12,29)(13,34,15,35)(14,33,16,36)(17,40,19,37)(18,39,20,38)$
4D-1 $4^{10}$ $5$ $4$ $30$ $( 1,23, 3,22)( 2,24, 4,21)( 5,25, 8,27)( 6,26, 7,28)( 9,30,11,31)(10,29,12,32)(13,35,15,34)(14,36,16,33)(17,37,19,40)(18,38,20,39)$
4E $4^{4},2^{12}$ $20$ $4$ $24$ $( 1,33)( 2,34)( 3,36)( 4,35)( 5,32, 6,31)( 7,30, 8,29)( 9,25,10,26)(11,27,12,28)(13,24)(14,23)(15,21)(16,22)(17,20)(18,19)(37,39)(38,40)$
4F $4^{4},2^{12}$ $20$ $4$ $24$ $( 1,36)( 2,35)( 3,33)( 4,34)( 5,29, 6,30)( 7,31, 8,32)( 9,27,10,28)(11,25,12,26)(13,21)(14,22)(15,24)(16,23)(17,18)(19,20)(37,38)(39,40)$
4G $4^{4},2^{12}$ $20$ $4$ $24$ $( 1, 4)( 2, 3)( 5,40, 6,39)( 7,38, 8,37)( 9,35)(10,36)(11,34)(12,33)(13,32)(14,31)(15,29)(16,30)(17,25,18,26)(19,27,20,28)(21,23)(22,24)$
4H $4^{4},2^{12}$ $20$ $4$ $24$ $( 1, 2)( 3, 4)( 5,37, 6,38)( 7,39, 8,40)( 9,34)(10,33)(11,35)(12,36)(13,29)(14,30)(15,32)(16,31)(17,27,18,28)(19,25,20,26)(21,22)(23,24)$
4I $4^{8},2^{4}$ $20$ $4$ $28$ $( 1, 3)( 2, 4)( 5,39, 6,40)( 7,37, 8,38)( 9,35,10,36)(11,34,12,33)(13,31,14,32)(15,30,16,29)(17,26,18,25)(19,28,20,27)(21,24)(22,23)$
4J $4^{8},1^{8}$ $20$ $4$ $24$ $( 5,38, 6,37)( 7,40, 8,39)( 9,34,10,33)(11,35,12,36)(13,30,14,29)(15,31,16,32)(17,28,18,27)(19,26,20,25)$
4K1 $4^{10}$ $20$ $4$ $30$ $( 1,15, 3,13)( 2,16, 4,14)( 5,11, 8, 9)( 6,12, 7,10)(17,37,19,40)(18,38,20,39)(21,36,24,33)(22,35,23,34)(25,31,27,30)(26,32,28,29)$
4K-1 $4^{10}$ $20$ $4$ $30$ $( 1,13, 3,15)( 2,14, 4,16)( 5, 9, 8,11)( 6,10, 7,12)(17,40,19,37)(18,39,20,38)(21,33,24,36)(22,34,23,35)(25,30,27,31)(26,29,28,32)$
4L1 $4^{10}$ $20$ $4$ $30$ $( 1,16, 3,14)( 2,15, 4,13)( 5,11, 7,10)( 6,12, 8, 9)(17,38,19,39)(18,37,20,40)(21,35,24,34)(22,36,23,33)(25,31,28,29)(26,32,27,30)$
4L-1 $4^{10}$ $20$ $4$ $30$ $( 1,14, 3,16)( 2,13, 4,15)( 5, 9, 7,12)( 6,10, 8,11)(17,39,19,38)(18,40,20,37)(21,34,24,35)(22,33,23,36)(25,30,28,32)(26,29,27,31)$
4M1 $4^{10}$ $20$ $4$ $30$ $( 1,24, 3,21)( 2,23, 4,22)( 5,19, 7,18)( 6,20, 8,17)( 9,13,11,15)(10,14,12,16)(25,39,28,37)(26,40,27,38)(29,35,32,34)(30,36,31,33)$
4M-1 $4^{10}$ $20$ $4$ $30$ $( 1,21, 3,24)( 2,22, 4,23)( 5,17, 7,20)( 6,18, 8,19)( 9,15,11,13)(10,16,12,14)(25,38,28,40)(26,37,27,39)(29,34,32,35)(30,33,31,36)$
4N1 $4^{10}$ $20$ $4$ $30$ $( 1,23, 3,22)( 2,24, 4,21)( 5,20, 7,17)( 6,19, 8,18)( 9,13,12,16)(10,14,11,15)(25,40,28,38)(26,39,27,37)(29,35,31,33)(30,36,32,34)$
4N-1 $4^{10}$ $20$ $4$ $30$ $( 1,22, 3,23)( 2,21, 4,24)( 5,18, 7,19)( 6,17, 8,20)( 9,15,12,14)(10,16,11,13)(25,37,28,39)(26,38,27,40)(29,34,31,36)(30,33,32,35)$
5A1 $5^{8}$ $32$ $5$ $32$ $( 1,36,25,20,12)( 2,35,26,19,11)( 3,33,27,18,10)( 4,34,28,17, 9)( 5,37,29,21,13)( 6,38,30,22,14)( 7,39,31,23,16)( 8,40,32,24,15)$
5A2 $5^{8}$ $32$ $5$ $32$ $( 1,25,12,36,20)( 2,26,11,35,19)( 3,27,10,33,18)( 4,28, 9,34,17)( 5,29,13,37,21)( 6,30,14,38,22)( 7,31,16,39,23)( 8,32,15,40,24)$
10A1 $10^{4}$ $32$ $10$ $36$ $( 1,18,36,10,25, 3,20,33,12,27)( 2,17,35, 9,26, 4,19,34,11,28)( 5,24,37,15,29, 8,21,40,13,32)( 6,23,38,16,30, 7,22,39,14,31)$
10A3 $10^{4}$ $32$ $10$ $36$ $( 1,10,20,27,36, 3,12,18,25,33)( 2, 9,19,28,35, 4,11,17,26,34)( 5,15,21,32,37, 8,13,24,29,40)( 6,16,22,31,38, 7,14,23,30,39)$
20A1 $20^{2}$ $32$ $20$ $38$ $( 1,30,18, 7,36,22,10,39,25,14, 3,31,20, 6,33,23,12,38,27,16)( 2,29,17, 8,35,21, 9,40,26,13, 4,32,19, 5,34,24,11,37,28,15)$
20A-1 $20^{2}$ $32$ $20$ $38$ $( 1,16,27,38,12,23,33, 6,20,31, 3,14,25,39,10,22,36, 7,18,30)( 2,15,28,37,11,24,34, 5,19,32, 4,13,26,40, 9,21,35, 8,17,29)$
20A3 $20^{2}$ $32$ $20$ $38$ $( 1, 7,10,14,20,23,27,30,36,39, 3, 6,12,16,18,22,25,31,33,38)( 2, 8, 9,13,19,24,28,29,35,40, 4, 5,11,15,17,21,26,32,34,37)$
20A-3 $20^{2}$ $32$ $20$ $38$ $( 1,38,33,31,25,22,18,16,12, 6, 3,39,36,30,27,23,20,14,10, 7)( 2,37,34,32,26,21,17,15,11, 5, 4,40,35,29,28,24,19,13, 9, 8)$

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