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G = C41⋊C5order 205 = 5·41

The semidirect product of C41 and C5 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary

Aliases: C41⋊C5, SmallGroup(205,1)

Series: Derived Chief Lower central Upper central

C1C41 — C41⋊C5
C1C41 — C41⋊C5
C41 — C41⋊C5
C1

Generators and relations for C41⋊C5
 G = < a,b | a41=b5=1, bab-1=a37 >

41C5

Character table of C41⋊C5

 class 15A5B5C5D41A41B41C41D41E41F41G41H
 size 14141414155555555
ρ11111111111111    trivial
ρ21ζ5ζ53ζ52ζ5411111111    linear of order 5
ρ31ζ54ζ52ζ53ζ511111111    linear of order 5
ρ41ζ52ζ5ζ54ζ5311111111    linear of order 5
ρ51ζ53ζ54ζ5ζ5211111111    linear of order 5
ρ650000ζ413741184116411041ζ41384134412841124111ζ41354127412441224115ζ413041294113417413ζ4136413341324120412ζ4140413141254123414ζ4126411941174114416ζ41394121419418415    complex faithful
ρ750000ζ41354127412441224115ζ413741184116411041ζ4136413341324120412ζ4140413141254123414ζ413041294113417413ζ4126411941174114416ζ41394121419418415ζ41384134412841124111    complex faithful
ρ850000ζ41384134412841124111ζ41394121419418415ζ413741184116411041ζ4136413341324120412ζ41354127412441224115ζ413041294113417413ζ4140413141254123414ζ4126411941174114416    complex faithful
ρ950000ζ4140413141254123414ζ413041294113417413ζ4126411941174114416ζ41384134412841124111ζ41394121419418415ζ413741184116411041ζ41354127412441224115ζ4136413341324120412    complex faithful
ρ1050000ζ4126411941174114416ζ4140413141254123414ζ41394121419418415ζ413741184116411041ζ41384134412841124111ζ41354127412441224115ζ4136413341324120412ζ413041294113417413    complex faithful
ρ1150000ζ413041294113417413ζ4136413341324120412ζ4140413141254123414ζ41394121419418415ζ4126411941174114416ζ41384134412841124111ζ413741184116411041ζ41354127412441224115    complex faithful
ρ1250000ζ41394121419418415ζ4126411941174114416ζ41384134412841124111ζ41354127412441224115ζ413741184116411041ζ4136413341324120412ζ413041294113417413ζ4140413141254123414    complex faithful
ρ1350000ζ4136413341324120412ζ41354127412441224115ζ413041294113417413ζ4126411941174114416ζ4140413141254123414ζ41394121419418415ζ41384134412841124111ζ413741184116411041    complex faithful

Smallest permutation representation of C41⋊C5
On 41 points: primitive
Generators in S41
(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)
(2 11 19 17 38)(3 21 37 33 34)(4 31 14 8 30)(5 41 32 24 26)(6 10 9 40 22)(7 20 27 15 18)(12 29 35 13 39)(16 28 25 36 23)

G:=sub<Sym(41)| (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), (2,11,19,17,38)(3,21,37,33,34)(4,31,14,8,30)(5,41,32,24,26)(6,10,9,40,22)(7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)>;

G:=Group( (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), (2,11,19,17,38)(3,21,37,33,34)(4,31,14,8,30)(5,41,32,24,26)(6,10,9,40,22)(7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23) );

G=PermutationGroup([(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)], [(2,11,19,17,38),(3,21,37,33,34),(4,31,14,8,30),(5,41,32,24,26),(6,10,9,40,22),(7,20,27,15,18),(12,29,35,13,39),(16,28,25,36,23)])

C41⋊C5 is a maximal subgroup of   C41⋊C10

Matrix representation of C41⋊C5 in GL5(𝔽821)

01000
00100
00010
00001
1408787471550
,
10000
313753518596433
57778050332239
23980732619802
545752453594587

G:=sub<GL(5,GF(821))| [0,0,0,0,1,1,0,0,0,408,0,1,0,0,787,0,0,1,0,471,0,0,0,1,550],[1,313,577,239,545,0,753,780,807,752,0,518,503,32,453,0,596,322,619,594,0,433,39,802,587] >;

C41⋊C5 in GAP, Magma, Sage, TeX

C_{41}\rtimes C_5
% in TeX

G:=Group("C41:C5");
// GroupNames label

G:=SmallGroup(205,1);
// by ID

G=gap.SmallGroup(205,1);
# by ID

G:=PCGroup([2,-5,-41,201]);
// Polycyclic

G:=Group<a,b|a^41=b^5=1,b*a*b^-1=a^37>;
// generators/relations

Export

Subgroup lattice of C41⋊C5 in TeX
Character table of C41⋊C5 in TeX

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