LMFDB - Galois group: 41T3

Show commands: Gap / Magma / Oscar / SageMath

comment:Define the Galois group

magma:G := TransitiveGroup(41, 3);

sage:G = TransitiveGroup(41, 3)

oscar:G = transitive_group(41, 3)

gap:G := TransitiveGroup(41, 3);

Group invariants

Abstract group:	$C_{41}:C_{4}$	comment:Abstract group ID magma:IdentifyGroup(G); sage:G.id() oscar:small_group_identification(G) gap:IdGroup(G);
Order:	$164=2^{2} \cdot 41$	comment:Order magma:Order(G); sage:G.order() oscar:order(G) gap:Order(G);
Cyclic:	no	comment:Determine if group is cyclic magma:IsCyclic(G); sage:G.is_cyclic() oscar:is_cyclic(G) gap:IsCyclic(G);
Abelian:	no	comment:Determine if group is abelian magma:IsAbelian(G); sage:G.is_abelian() oscar:is_abelian(G) gap:IsAbelian(G);
Solvable:	yes	comment:Determine if group is solvable magma:IsSolvable(G); sage:G.is_solvable() oscar:is_solvable(G) gap:IsSolvable(G);
Nilpotency class:	not nilpotent	comment:Nilpotency class magma:NilpotencyClass(G); sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1 oscar:if is_nilpotent(G) nilpotency_class(G) end gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;

Group action invariants

Degree $n$:	$41$	comment:Degree magma:t, n := TransitiveGroupIdentification(G); n; sage:G.degree() oscar:degree(G) gap:NrMovedPoints(G);
Transitive number $t$:	$3$	comment:Transitive number magma:t, n := TransitiveGroupIdentification(G); t; sage:G.transitive_number() oscar:transitive_group_identification(G)[2] gap:TransitiveIdentification(G);
Parity:	$1$	comment:Parity magma:IsEven(G); sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup()) oscar:is_even(G) gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
Transitivity:	1
Primitive:	yes	comment:Determine if group is primitive magma:IsPrimitive(G); sage:G.is_primitive() oscar:is_primitive(G) gap:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$1$	comment:Order of the centralizer of G in S_n magma:Order(Centralizer(SymmetricGroup(n), G)); sage:SymmetricGroup(41).centralizer(G).order() oscar:order(centralizer(symmetric_group(41), G)[1]) gap:Order(Centralizer(SymmetricGroup(41), G));
Generators:	$(1,32,40,9)(2,23,39,18)(3,14,38,27)(4,5,37,36)(6,28,35,13)(7,19,34,22)(8,10,33,31)(11,24,30,17)(12,15,29,26)(16,20,25,21)$, $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$	comment:Generators magma:Generators(G); sage:G.gens() oscar:gens(G) gap:GeneratorsOfGroup(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$
$4$: $C_4$

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

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

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{41}$	$1$	$1$	$0$	$()$
2A	$2^{20},1$	$41$	$2$	$20$	$( 1, 6)( 2, 5)( 3, 4)( 7,41)( 8,40)( 9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)$
4A1	$4^{10},1$	$41$	$4$	$30$	$( 1,26, 6,22)( 2,17, 5,31)( 3, 8, 4,40)( 7,13,41,35)( 9,36,39,12)(10,27,38,21)(11,18,37,30)(14,32,34,16)(15,23,33,25)(19,28,29,20)$
4A-1	$4^{10},1$	$41$	$4$	$30$	$( 1,22, 6,26)( 2,31, 5,17)( 3,40, 4, 8)( 7,35,41,13)( 9,12,39,36)(10,21,38,27)(11,30,37,18)(14,16,34,32)(15,25,33,23)(19,20,29,28)$
41A1	$41$	$4$	$41$	$40$	$( 1,24, 6,29,11,34,16,39,21, 3,26, 8,31,13,36,18,41,23, 5,28,10,33,15,38,20, 2,25, 7,30,12,35,17,40,22, 4,27, 9,32,14,37,19)$
41A2	$41$	$4$	$41$	$40$	$( 1, 6,11,16,21,26,31,36,41, 5,10,15,20,25,30,35,40, 4, 9,14,19,24,29,34,39, 3, 8,13,18,23,28,33,38, 2, 7,12,17,22,27,32,37)$
41A3	$41$	$4$	$41$	$40$	$( 1,29,16, 3,31,18, 5,33,20, 7,35,22, 9,37,24,11,39,26,13,41,28,15, 2,30,17, 4,32,19, 6,34,21, 8,36,23,10,38,25,12,40,27,14)$
41A4	$41$	$4$	$41$	$40$	$( 1,11,21,31,41,10,20,30,40, 9,19,29,39, 8,18,28,38, 7,17,27,37, 6,16,26,36, 5,15,25,35, 4,14,24,34, 3,13,23,33, 2,12,22,32)$
41A6	$41$	$4$	$41$	$40$	$( 1,16,31, 5,20,35, 9,24,39,13,28, 2,17,32, 6,21,36,10,25,40,14,29, 3,18,33, 7,22,37,11,26,41,15,30, 4,19,34, 8,23,38,12,27)$
41A7	$41$	$4$	$41$	$40$	$( 1,39,36,33,30,27,24,21,18,15,12, 9, 6, 3,41,38,35,32,29,26,23,20,17,14,11, 8, 5, 2,40,37,34,31,28,25,22,19,16,13,10, 7, 4)$
41A8	$41$	$4$	$41$	$40$	$( 1,21,41,20,40,19,39,18,38,17,37,16,36,15,35,14,34,13,33,12,32,11,31,10,30, 9,29, 8,28, 7,27, 6,26, 5,25, 4,24, 3,23, 2,22)$
41A11	$41$	$4$	$41$	$40$	$( 1, 8,15,22,29,36, 2, 9,16,23,30,37, 3,10,17,24,31,38, 4,11,18,25,32,39, 5,12,19,26,33,40, 6,13,20,27,34,41, 7,14,21,28,35)$
41A12	$41$	$4$	$41$	$40$	$( 1,31,20, 9,39,28,17, 6,36,25,14, 3,33,22,11,41,30,19, 8,38,27,16, 5,35,24,13, 2,32,21,10,40,29,18, 7,37,26,15, 4,34,23,12)$
41A16	$41$	$4$	$41$	$40$	$( 1,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

Malle's constant $a(G)$: $1/20$

comment:Conjugacy classes

magma:ConjugacyClasses(G);

sage:G.conjugacy_classes()

oscar:conjugacy_classes(G)

gap:ConjugacyClasses(G);

Character table

Size

41A2

41A4

41A6

41A8

41A12

41A3

41A16

41A7

41A11

41A1

41 P

4A1

4A-1

41A4

41A8

41A12

41A16

41A11

41A6

41A1

41A3

41A7

41A2

Type

164.3.1a

1

1

1

1

1

1

1

1

1

1

1

1

1

1

164.3.1b

1

1

- 1

- 1

1

1

1

1

1

1

1

1

1

1

164.3.1c1

1

- 1

- i

i

1

1

1

1

1

1

1

1

1

1

164.3.1c2

1

- 1

i

- i

1

1

1

1

1

1

1

1

1

1

164.3.4a1

4

0

0

0

ζ_{41}^{- 20} + ζ_{41}^{- 16} + ζ_{41}^{16} + ζ_{41}^{20}

ζ_{41}^{- 19} + ζ_{41}^{- 7} + ζ_{41}^{7} + ζ_{41}^{19}

ζ_{41}^{- 17} + ζ_{41}^{- 11} + ζ_{41}^{11} + ζ_{41}^{17}

ζ_{41}^{- 9} + ζ_{41}^{- 1} + ζ_{41} + ζ_{41}^{9}

ζ_{41}^{- 18} + ζ_{41}^{- 2} + ζ_{41}^{2} + ζ_{41}^{18}

ζ_{41}^{- 5} + ζ_{41}^{- 4} + ζ_{41}^{4} + ζ_{41}^{5}

ζ_{41}^{- 13} + ζ_{41}^{- 6} + ζ_{41}^{6} + ζ_{41}^{13}

ζ_{41}^{- 10} + ζ_{41}^{- 8} + ζ_{41}^{8} + ζ_{41}^{10}

ζ_{41}^{- 15} + ζ_{41}^{- 12} + ζ_{41}^{12} + ζ_{41}^{15}

ζ_{41}^{- 14} + ζ_{41}^{- 3} + ζ_{41}^{3} + ζ_{41}^{14}

164.3.4a2

4

0

0

0

ζ_{41}^{- 17} + ζ_{41}^{- 11} + ζ_{41}^{11} + ζ_{41}^{17}

ζ_{41}^{- 10} + ζ_{41}^{- 8} + ζ_{41}^{8} + ζ_{41}^{10}

ζ_{41}^{- 5} + ζ_{41}^{- 4} + ζ_{41}^{4} + ζ_{41}^{5}

ζ_{41}^{- 19} + ζ_{41}^{- 7} + ζ_{41}^{7} + ζ_{41}^{19}

ζ_{41}^{- 14} + ζ_{41}^{- 3} + ζ_{41}^{3} + ζ_{41}^{14}

ζ_{41}^{- 13} + ζ_{41}^{- 6} + ζ_{41}^{6} + ζ_{41}^{13}

ζ_{41}^{- 9} + ζ_{41}^{- 1} + ζ_{41} + ζ_{41}^{9}

ζ_{41}^{- 15} + ζ_{41}^{- 12} + ζ_{41}^{12} + ζ_{41}^{15}

ζ_{41}^{- 18} + ζ_{41}^{- 2} + ζ_{41}^{2} + ζ_{41}^{18}

ζ_{41}^{- 20} + ζ_{41}^{- 16} + ζ_{41}^{16} + ζ_{41}^{20}

164.3.4a3

4

0

0

0

ζ_{41}^{- 15} + ζ_{41}^{- 12} + ζ_{41}^{12} + ζ_{41}^{15}

ζ_{41}^{- 5} + ζ_{41}^{- 4} + ζ_{41}^{4} + ζ_{41}^{5}

ζ_{41}^{- 18} + ζ_{41}^{- 2} + ζ_{41}^{2} + ζ_{41}^{18}

ζ_{41}^{- 17} + ζ_{41}^{- 11} + ζ_{41}^{11} + ζ_{41}^{17}

ζ_{41}^{- 19} + ζ_{41}^{- 7} + ζ_{41}^{7} + ζ_{41}^{19}

ζ_{41}^{- 14} + ζ_{41}^{- 3} + ζ_{41}^{3} + ζ_{41}^{14}

ζ_{41}^{- 20} + ζ_{41}^{- 16} + ζ_{41}^{16} + ζ_{41}^{20}

ζ_{41}^{- 13} + ζ_{41}^{- 6} + ζ_{41}^{6} + ζ_{41}^{13}

ζ_{41}^{- 9} + ζ_{41}^{- 1} + ζ_{41} + ζ_{41}^{9}

ζ_{41}^{- 10} + ζ_{41}^{- 8} + ζ_{41}^{8} + ζ_{41}^{10}

164.3.4a4

4

0

0

0

ζ_{41}^{- 19} + ζ_{41}^{- 7} + ζ_{41}^{7} + ζ_{41}^{19}

ζ_{41}^{- 20} + ζ_{41}^{- 16} + ζ_{41}^{16} + ζ_{41}^{20}

ζ_{41}^{- 10} + ζ_{41}^{- 8} + ζ_{41}^{8} + ζ_{41}^{10}

ζ_{41}^{- 14} + ζ_{41}^{- 3} + ζ_{41}^{3} + ζ_{41}^{14}

ζ_{41}^{- 13} + ζ_{41}^{- 6} + ζ_{41}^{6} + ζ_{41}^{13}

ζ_{41}^{- 15} + ζ_{41}^{- 12} + ζ_{41}^{12} + ζ_{41}^{15}

ζ_{41}^{- 18} + ζ_{41}^{- 2} + ζ_{41}^{2} + ζ_{41}^{18}

ζ_{41}^{- 17} + ζ_{41}^{- 11} + ζ_{41}^{11} + ζ_{41}^{17}

ζ_{41}^{- 5} + ζ_{41}^{- 4} + ζ_{41}^{4} + ζ_{41}^{5}

ζ_{41}^{- 9} + ζ_{41}^{- 1} + ζ_{41} + ζ_{41}^{9}

164.3.4a5

4

0

0

0

ζ_{41}^{- 18} + ζ_{41}^{- 2} + ζ_{41}^{2} + ζ_{41}^{18}

ζ_{41}^{- 13} + ζ_{41}^{- 6} + ζ_{41}^{6} + ζ_{41}^{13}

ζ_{41}^{- 14} + ζ_{41}^{- 3} + ζ_{41}^{3} + ζ_{41}^{14}

ζ_{41}^{- 5} + ζ_{41}^{- 4} + ζ_{41}^{4} + ζ_{41}^{5}

ζ_{41}^{- 10} + ζ_{41}^{- 8} + ζ_{41}^{8} + ζ_{41}^{10}

ζ_{41}^{- 20} + ζ_{41}^{- 16} + ζ_{41}^{16} + ζ_{41}^{20}

ζ_{41}^{- 17} + ζ_{41}^{- 11} + ζ_{41}^{11} + ζ_{41}^{17}

ζ_{41}^{- 9} + ζ_{41}^{- 1} + ζ_{41} + ζ_{41}^{9}

ζ_{41}^{- 19} + ζ_{41}^{- 7} + ζ_{41}^{7} + ζ_{41}^{19}

ζ_{41}^{- 15} + ζ_{41}^{- 12} + ζ_{41}^{12} + ζ_{41}^{15}

164.3.4a6

4

0

0

0

ζ_{41}^{- 13} + ζ_{41}^{- 6} + ζ_{41}^{6} + ζ_{41}^{13}

ζ_{41}^{- 18} + ζ_{41}^{- 2} + ζ_{41}^{2} + ζ_{41}^{18}

ζ_{41}^{- 9} + ζ_{41}^{- 1} + ζ_{41} + ζ_{41}^{9}

ζ_{41}^{- 15} + ζ_{41}^{- 12} + ζ_{41}^{12} + ζ_{41}^{15}

ζ_{41}^{- 17} + ζ_{41}^{- 11} + ζ_{41}^{11} + ζ_{41}^{17}

ζ_{41}^{- 19} + ζ_{41}^{- 7} + ζ_{41}^{7} + ζ_{41}^{19}

ζ_{41}^{- 10} + ζ_{41}^{- 8} + ζ_{41}^{8} + ζ_{41}^{10}

ζ_{41}^{- 14} + ζ_{41}^{- 3} + ζ_{41}^{3} + ζ_{41}^{14}

ζ_{41}^{- 20} + ζ_{41}^{- 16} + ζ_{41}^{16} + ζ_{41}^{20}

ζ_{41}^{- 5} + ζ_{41}^{- 4} + ζ_{41}^{4} + ζ_{41}^{5}

164.3.4a7

4

0

0

0

ζ_{41}^{- 14} + ζ_{41}^{- 3} + ζ_{41}^{3} + ζ_{41}^{14}

ζ_{41}^{- 9} + ζ_{41}^{- 1} + ζ_{41} + ζ_{41}^{9}

ζ_{41}^{- 20} + ζ_{41}^{- 16} + ζ_{41}^{16} + ζ_{41}^{20}

ζ_{41}^{- 13} + ζ_{41}^{- 6} + ζ_{41}^{6} + ζ_{41}^{13}

ζ_{41}^{- 15} + ζ_{41}^{- 12} + ζ_{41}^{12} + ζ_{41}^{15}

ζ_{41}^{- 17} + ζ_{41}^{- 11} + ζ_{41}^{11} + ζ_{41}^{17}

ζ_{41}^{- 5} + ζ_{41}^{- 4} + ζ_{41}^{4} + ζ_{41}^{5}

ζ_{41}^{- 19} + ζ_{41}^{- 7} + ζ_{41}^{7} + ζ_{41}^{19}

ζ_{41}^{- 10} + ζ_{41}^{- 8} + ζ_{41}^{8} + ζ_{41}^{10}

ζ_{41}^{- 18} + ζ_{41}^{- 2} + ζ_{41}^{2} + ζ_{41}^{18}

164.3.4a8

4

0

0

0

ζ_{41}^{- 10} + ζ_{41}^{- 8} + ζ_{41}^{8} + ζ_{41}^{10}

ζ_{41}^{- 17} + ζ_{41}^{- 11} + ζ_{41}^{11} + ζ_{41}^{17}

ζ_{41}^{- 15} + ζ_{41}^{- 12} + ζ_{41}^{12} + ζ_{41}^{15}

ζ_{41}^{- 20} + ζ_{41}^{- 16} + ζ_{41}^{16} + ζ_{41}^{20}

ζ_{41}^{- 9} + ζ_{41}^{- 1} + ζ_{41} + ζ_{41}^{9}

ζ_{41}^{- 18} + ζ_{41}^{- 2} + ζ_{41}^{2} + ζ_{41}^{18}

ζ_{41}^{- 14} + ζ_{41}^{- 3} + ζ_{41}^{3} + ζ_{41}^{14}

ζ_{41}^{- 5} + ζ_{41}^{- 4} + ζ_{41}^{4} + ζ_{41}^{5}

ζ_{41}^{- 13} + ζ_{41}^{- 6} + ζ_{41}^{6} + ζ_{41}^{13}

ζ_{41}^{- 19} + ζ_{41}^{- 7} + ζ_{41}^{7} + ζ_{41}^{19}

164.3.4a9

4

0

0

0

ζ_{41}^{- 9} + ζ_{41}^{- 1} + ζ_{41} + ζ_{41}^{9}

ζ_{41}^{- 14} + ζ_{41}^{- 3} + ζ_{41}^{3} + ζ_{41}^{14}

ζ_{41}^{- 19} + ζ_{41}^{- 7} + ζ_{41}^{7} + ζ_{41}^{19}

ζ_{41}^{- 18} + ζ_{41}^{- 2} + ζ_{41}^{2} + ζ_{41}^{18}

ζ_{41}^{- 5} + ζ_{41}^{- 4} + ζ_{41}^{4} + ζ_{41}^{5}

ζ_{41}^{- 10} + ζ_{41}^{- 8} + ζ_{41}^{8} + ζ_{41}^{10}

ζ_{41}^{- 15} + ζ_{41}^{- 12} + ζ_{41}^{12} + ζ_{41}^{15}

ζ_{41}^{- 20} + ζ_{41}^{- 16} + ζ_{41}^{16} + ζ_{41}^{20}

ζ_{41}^{- 17} + ζ_{41}^{- 11} + ζ_{41}^{11} + ζ_{41}^{17}

ζ_{41}^{- 13} + ζ_{41}^{- 6} + ζ_{41}^{6} + ζ_{41}^{13}

164.3.4a10

4

0

0

0

ζ_{41}^{- 5} + ζ_{41}^{- 4} + ζ_{41}^{4} + ζ_{41}^{5}

ζ_{41}^{- 15} + ζ_{41}^{- 12} + ζ_{41}^{12} + ζ_{41}^{15}

ζ_{41}^{- 13} + ζ_{41}^{- 6} + ζ_{41}^{6} + ζ_{41}^{13}

ζ_{41}^{- 10} + ζ_{41}^{- 8} + ζ_{41}^{8} + ζ_{41}^{10}

ζ_{41}^{- 20} + ζ_{41}^{- 16} + ζ_{41}^{16} + ζ_{41}^{20}

ζ_{41}^{- 9} + ζ_{41}^{- 1} + ζ_{41} + ζ_{41}^{9}

ζ_{41}^{- 19} + ζ_{41}^{- 7} + ζ_{41}^{7} + ζ_{41}^{19}

ζ_{41}^{- 18} + ζ_{41}^{- 2} + ζ_{41}^{2} + ζ_{41}^{18}

ζ_{41}^{- 14} + ζ_{41}^{- 3} + ζ_{41}^{3} + ζ_{41}^{14}

ζ_{41}^{- 17} + ζ_{41}^{- 11} + ζ_{41}^{11} + ζ_{41}^{17}

comment:Character table

magma:CharacterTable(G);

sage:G.character_table()

oscar:character_table(G)

gap:CharacterTable(G);

Regular extensions

Data not computed

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Low degree siblings

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Character table

Regular extensions

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	41T3

Degree	$41$
Order	$164$
Cyclic	no
Abelian	no
Solvable	yes
Transitivity	$1$
Primitive	yes
$p$-group	no
Group:	$C_{41}:C_{4}$