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Properties

Label 20T1
20T1 1 4 1->4 12 1->12 2 3 2->3 11 2->11 5 3->5 14 3->14 6 4->6 13 4->13 7 5->7 15 5->15 8 6->8 16 6->16 10 7->10 17 7->17 9 8->9 18 8->18 9->12 20 9->20 10->11 19 10->19 11->1 11->14 12->2 12->13 13->3 13->16 14->4 14->15 15->6 15->17 16->5 16->18 17->8 17->19 18->7 18->20 19->1 19->9 20->2 20->10
Degree $20$
Order $20$
Cyclic yes
Abelian yes
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{20}$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 1);
 
Copy content sage:G = TransitiveGroup(20, 1)
 
Copy content oscar:G = transitive_group(20, 1)
 
Copy content gap:G := TransitiveGroup(20, 1);
 

Group invariants

Abstract group:  $C_{20}$
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:  $20=2^{2} \cdot 5$
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:  yes
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:  yes
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:  $1$
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 $n$:  $20$
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 $t$:  $1$
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);
 
$\card{\Aut(F/K)}$:  $20$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(20).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(20), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(20), G));
 
Generators:  $(1,12,2,11)(3,14,4,13)(5,15,6,16)(7,17,8,18)(9,20,10,19)$, $(1,4,6,8,9,12,13,16,18,20,2,3,5,7,10,11,14,15,17,19)$
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)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$10$:  $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $C_5$

Degree 10: $C_{10}$

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 $1^{20}$ $1$ $1$ $0$ $()$
2A $2^{10}$ $1$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
4A1 $4^{5}$ $1$ $4$ $15$ $( 1,12, 2,11)( 3,14, 4,13)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$
4A-1 $4^{5}$ $1$ $4$ $15$ $( 1,11, 2,12)( 3,13, 4,14)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)$
5A1 $5^{4}$ $1$ $5$ $16$ $( 1,14, 5,18, 9)( 2,13, 6,17,10)( 3,16, 8,19,11)( 4,15, 7,20,12)$
5A-1 $5^{4}$ $1$ $5$ $16$ $( 1, 9,18, 5,14)( 2,10,17, 6,13)( 3,11,19, 8,16)( 4,12,20, 7,15)$
5A2 $5^{4}$ $1$ $5$ $16$ $( 1, 5, 9,14,18)( 2, 6,10,13,17)( 3, 8,11,16,19)( 4, 7,12,15,20)$
5A-2 $5^{4}$ $1$ $5$ $16$ $( 1,18,14, 9, 5)( 2,17,13,10, 6)( 3,19,16,11, 8)( 4,20,15,12, 7)$
10A1 $10^{2}$ $1$ $10$ $18$ $( 1,17,14,10, 5, 2,18,13, 9, 6)( 3,20,16,12, 8, 4,19,15,11, 7)$
10A-1 $10^{2}$ $1$ $10$ $18$ $( 1, 6, 9,13,18, 2, 5,10,14,17)( 3, 7,11,15,19, 4, 8,12,16,20)$
10A3 $10^{2}$ $1$ $10$ $18$ $( 1,10,18, 6,14, 2, 9,17, 5,13)( 3,12,19, 7,16, 4,11,20, 8,15)$
10A-3 $10^{2}$ $1$ $10$ $18$ $( 1,13, 5,17, 9, 2,14, 6,18,10)( 3,15, 8,20,11, 4,16, 7,19,12)$
20A1 $20$ $1$ $20$ $19$ $( 1,20,17,16,14,12,10, 8, 5, 4, 2,19,18,15,13,11, 9, 7, 6, 3)$
20A-1 $20$ $1$ $20$ $19$ $( 1, 3, 6, 7, 9,11,13,15,18,19, 2, 4, 5, 8,10,12,14,16,17,20)$
20A3 $20$ $1$ $20$ $19$ $( 1,16,10, 4,18,11, 6,20,14, 8, 2,15, 9, 3,17,12, 5,19,13, 7)$
20A-3 $20$ $1$ $20$ $19$ $( 1, 7,13,19, 5,12,17, 3, 9,15, 2, 8,14,20, 6,11,18, 4,10,16)$
20A7 $20$ $1$ $20$ $19$ $( 1, 8,13,20, 5,11,17, 4, 9,16, 2, 7,14,19, 6,12,18, 3,10,15)$
20A-7 $20$ $1$ $20$ $19$ $( 1,15,10, 3,18,12, 6,19,14, 7, 2,16, 9, 4,17,11, 5,20,13, 8)$
20A9 $20$ $1$ $20$ $19$ $( 1, 4, 6, 8, 9,12,13,16,18,20, 2, 3, 5, 7,10,11,14,15,17,19)$
20A-9 $20$ $1$ $20$ $19$ $( 1,19,17,15,14,11,10, 7, 5, 3, 2,20,18,16,13,12, 9, 8, 6, 4)$

Malle's constant $a(G)$:     $1/10$

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
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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
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
Type
20.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
20.2.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
20.2.1c1 C 1 −1 −i i 1 1 1 1 −1 −1 −1 −1 −i i i −i i −i −i i
20.2.1c2 C 1 −1 i −i 1 1 1 1 −1 −1 −1 −1 i −i −i i −i i i −i
20.2.1d1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52
20.2.1d2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2
20.2.1d3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5
20.2.1d4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1
20.2.1e1 C 1 1 −1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−1 −ζ5 −ζ5−2 −ζ52
20.2.1e2 C 1 1 −1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5 −ζ5−1 −ζ52 −ζ5−2
20.2.1e3 C 1 1 −1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ52 −ζ5−2 −ζ5−1 −ζ5
20.2.1e4 C 1 1 −1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5−2 −ζ52 −ζ5 −ζ5−1
20.2.1f1 C 1 −1 −ζ205 ζ205 −ζ202 ζ208 ζ204 −ζ206 ζ206 −ζ204 −ζ208 ζ202 ζ203 −ζ207 ζ209 −ζ20 ζ20 −ζ209 ζ207 −ζ203
20.2.1f2 C 1 −1 ζ205 −ζ205 ζ208 −ζ202 −ζ206 ζ204 −ζ204 ζ206 ζ202 −ζ208 −ζ207 ζ203 −ζ20 ζ209 −ζ209 ζ20 −ζ203 ζ207
20.2.1f3 C 1 −1 −ζ205 ζ205 ζ208 −ζ202 −ζ206 ζ204 −ζ204 ζ206 ζ202 −ζ208 ζ207 −ζ203 ζ20 −ζ209 ζ209 −ζ20 ζ203 −ζ207
20.2.1f4 C 1 −1 ζ205 −ζ205 −ζ202 ζ208 ζ204 −ζ206 ζ206 −ζ204 −ζ208 ζ202 −ζ203 ζ207 −ζ209 ζ20 −ζ20 ζ209 −ζ207 ζ203
20.2.1f5 C 1 −1 −ζ205 ζ205 −ζ206 ζ204 −ζ202 ζ208 −ζ208 ζ202 −ζ204 ζ206 −ζ209 ζ20 −ζ207 ζ203 −ζ203 ζ207 −ζ20 ζ209
20.2.1f6 C 1 −1 ζ205 −ζ205 ζ204 −ζ206 ζ208 −ζ202 ζ202 −ζ208 ζ206 −ζ204 ζ20 −ζ209 ζ203 −ζ207 ζ207 −ζ203 ζ209 −ζ20
20.2.1f7 C 1 −1 −ζ205 ζ205 ζ204 −ζ206 ζ208 −ζ202 ζ202 −ζ208 ζ206 −ζ204 −ζ20 ζ209 −ζ203 ζ207 −ζ207 ζ203 −ζ209 ζ20
20.2.1f8 C 1 −1 ζ205 −ζ205 −ζ206 ζ204 −ζ202 ζ208 −ζ208 ζ202 −ζ204 ζ206 ζ209 −ζ20 ζ207 −ζ203 ζ203 −ζ207 ζ20 −ζ209

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

$f_{ 1 } =$ $x^{20} + \left(-80 t^{8} + 80 t^{6} - 80 t^{4} + 80 t^{2} - 80\right) x^{18} + \left(1120 t^{16} + 2960 t^{14} - 9840 t^{12} + 11520 t^{10} - 9600 t^{8} + 10720 t^{6} - 6640 t^{4} - 240 t^{2} + 1920\right) x^{16} + \left(-6400 t^{24} - 68800 t^{22} + 153600 t^{20} - 130000 t^{18} + 272000 t^{16} - 756800 t^{14} + 1108400 t^{12} - 1140800 t^{10} + 1150000 t^{8} - 1008000 t^{6} + 529600 t^{4} - 102800 t^{2} - 14400\right) x^{14} + \left(16640 t^{32} + 490240 t^{30} + 15200 t^{28} - 4833600 t^{26} + 14164000 t^{24} - 29956480 t^{22} + 57623520 t^{20} - 88589600 t^{18} + 102312800 t^{16} - 94692000 t^{14} + 75317120 t^{12} - 46843680 t^{10} + 15395200 t^{8} + 3146400 t^{6} - 4856000 t^{4} + 1290240 t^{2} + 23040\right) x^{12} + \left(-16384 t^{40} - 1479680 t^{38} - 8712960 t^{36} + 57463040 t^{34} - 131433920 t^{32} + 196921024 t^{30} - 278040640 t^{28} + 405555520 t^{26} - 618548480 t^{24} + 1096285440 t^{22} - 1929461824 t^{20} + 2781405760 t^{18} - 3219494080 t^{16} + 3147913920 t^{14} - 2617783360 t^{12} + 1716799744 t^{10} - 785233280 t^{8} + 209437440 t^{6} - 19274560 t^{4} - 2302720 t^{2} - 1024\right) x^{10} + \left(1638400 t^{46} + 42739200 t^{44} - 147065600 t^{42} - 303072000 t^{40} + 2961952000 t^{38} - 9999814400 t^{36} + 23957012800 t^{34} - 47982094400 t^{32} + 83879704000 t^{30} - 129117704000 t^{28} + 175952246400 t^{26} - 212104596800 t^{24} + 224386710400 t^{22} - 206729776000 t^{20} + 164773936000 t^{18} - 111578510400 t^{16} + 61318524800 t^{14} - 25058158400 t^{12} + 6367448000 t^{10} - 426408000 t^{8} - 252209600 t^{6} + 57395200 t^{4} + 102400 t^{2}\right) x^{8} + \left(-61440000 t^{52} - 256640000 t^{50} + 4579200000 t^{48} - 20783200000 t^{46} + 52267840000 t^{44} - 85874400000 t^{42} + 90832800000 t^{40} - 17090720000 t^{38} - 210501920000 t^{36} + 654056160000 t^{34} - 1278972960000 t^{32} + 1927875840000 t^{30} - 2396920160000 t^{28} + 2529641440000 t^{26} - 2276085760000 t^{24} + 1726327520000 t^{22} - 1079533280000 t^{20} + 533452640000 t^{18} - 186592160000 t^{16} + 27723360000 t^{14} + 13887520000 t^{12} - 10719680000 t^{10} + 3102880000 t^{8} - 352320000 t^{6} - 2560000 t^{4}\right) x^{6} + \left(1024000000 t^{58} - 7324000000 t^{56} + 8848000000 t^{54} + 103432000000 t^{52} - 662496000000 t^{50} + 2204260000000 t^{48} - 5166024000000 t^{46} + 9462136000000 t^{44} - 14207016000000 t^{42} + 17810468000000 t^{40} - 18663296000000 t^{38} + 16153068000000 t^{36} - 11100600000000 t^{34} + 5233860000000 t^{32} - 344040000000 t^{30} - 2397428000000 t^{28} + 2912128000000 t^{26} - 2096816000000 t^{24} + 1022496000000 t^{22} - 264868000000 t^{20} - 90248000000 t^{18} + 160376000000 t^{16} - 102928000000 t^{14} + 38188000000 t^{12} - 7864000000 t^{10} + 648000000 t^{8} + 16000000 t^{6}\right) x^{4} + \left(-6400000000 t^{64} + 91200000000 t^{62} - 606800000000 t^{60} + 2505200000000 t^{58} - 7200352000000 t^{56} + 15341024000000 t^{54} - 25217760000000 t^{52} + 32816976000000 t^{50} - 34047648000000 t^{48} + 27060976000000 t^{46} - 13085392000000 t^{44} - 3728400000000 t^{42} + 17519792000000 t^{40} - 23682576000000 t^{38} + 21654720000000 t^{36} - 14562800000000 t^{34} + 6452160000000 t^{32} - 235520000000 t^{30} - 2785440000000 t^{28} + 2889904000000 t^{26} - 1526928000000 t^{24} + 236880000000 t^{22} + 308288000000 t^{20} - 295744000000 t^{18} + 136512000000 t^{16} - 36944000000 t^{14} + 5280000000 t^{12} - 176000000 t^{10} - 32000000 t^{8}\right) x^{2} + \left(1936000000 t^{62} - 29744000000 t^{60} + 214080000000 t^{58} - 958080000000 t^{56} + 2983520000000 t^{54} - 6851824000000 t^{52} + 11986896000000 t^{50} - 16218080000000 t^{48} + 16882000000000 t^{46} - 12824720000000 t^{44} + 5381648000000 t^{42} + 2445168000000 t^{40} - 7633840000000 t^{38} + 8588160000000 t^{36} - 5927760000000 t^{34} + 2034768000000 t^{32} + 695888000000 t^{30} - 1455920000000 t^{28} + 990400000000 t^{26} - 359280000000 t^{24} + 39216000000 t^{22} + 27376000000 t^{20} - 14240000000 t^{18} + 2320000000 t^{16} + 240000000 t^{14} - 144000000 t^{12} + 16000000 t^{10}\right)$ Copy content Toggle raw display