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## G = C41⋊C5order 205 = 5·41

### The semidirect product of C41 and C5 acting faithfully

Aliases: C41⋊C5, SmallGroup(205,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C41 — C41⋊C5
 Chief series C1 — C41 — C41⋊C5
 Lower central C41 — C41⋊C5
 Upper central C1

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

Character table of C41⋊C5

 class 1 5A 5B 5C 5D 41A 41B 41C 41D 41E 41F 41G 41H size 1 41 41 41 41 5 5 5 5 5 5 5 5 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ5 ζ53 ζ52 ζ54 1 1 1 1 1 1 1 1 linear of order 5 ρ3 1 ζ54 ζ52 ζ53 ζ5 1 1 1 1 1 1 1 1 linear of order 5 ρ4 1 ζ52 ζ5 ζ54 ζ53 1 1 1 1 1 1 1 1 linear of order 5 ρ5 1 ζ53 ζ54 ζ5 ζ52 1 1 1 1 1 1 1 1 linear of order 5 ρ6 5 0 0 0 0 ζ4137+ζ4118+ζ4116+ζ4110+ζ41 ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 ζ4130+ζ4129+ζ4113+ζ417+ζ413 ζ4136+ζ4133+ζ4132+ζ4120+ζ412 ζ4140+ζ4131+ζ4125+ζ4123+ζ414 ζ4126+ζ4119+ζ4117+ζ4114+ζ416 ζ4139+ζ4121+ζ419+ζ418+ζ415 complex faithful ρ7 5 0 0 0 0 ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 ζ4137+ζ4118+ζ4116+ζ4110+ζ41 ζ4136+ζ4133+ζ4132+ζ4120+ζ412 ζ4140+ζ4131+ζ4125+ζ4123+ζ414 ζ4130+ζ4129+ζ4113+ζ417+ζ413 ζ4126+ζ4119+ζ4117+ζ4114+ζ416 ζ4139+ζ4121+ζ419+ζ418+ζ415 ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 complex faithful ρ8 5 0 0 0 0 ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 ζ4139+ζ4121+ζ419+ζ418+ζ415 ζ4137+ζ4118+ζ4116+ζ4110+ζ41 ζ4136+ζ4133+ζ4132+ζ4120+ζ412 ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 ζ4130+ζ4129+ζ4113+ζ417+ζ413 ζ4140+ζ4131+ζ4125+ζ4123+ζ414 ζ4126+ζ4119+ζ4117+ζ4114+ζ416 complex faithful ρ9 5 0 0 0 0 ζ4140+ζ4131+ζ4125+ζ4123+ζ414 ζ4130+ζ4129+ζ4113+ζ417+ζ413 ζ4126+ζ4119+ζ4117+ζ4114+ζ416 ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 ζ4139+ζ4121+ζ419+ζ418+ζ415 ζ4137+ζ4118+ζ4116+ζ4110+ζ41 ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 ζ4136+ζ4133+ζ4132+ζ4120+ζ412 complex faithful ρ10 5 0 0 0 0 ζ4126+ζ4119+ζ4117+ζ4114+ζ416 ζ4140+ζ4131+ζ4125+ζ4123+ζ414 ζ4139+ζ4121+ζ419+ζ418+ζ415 ζ4137+ζ4118+ζ4116+ζ4110+ζ41 ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 ζ4136+ζ4133+ζ4132+ζ4120+ζ412 ζ4130+ζ4129+ζ4113+ζ417+ζ413 complex faithful ρ11 5 0 0 0 0 ζ4130+ζ4129+ζ4113+ζ417+ζ413 ζ4136+ζ4133+ζ4132+ζ4120+ζ412 ζ4140+ζ4131+ζ4125+ζ4123+ζ414 ζ4139+ζ4121+ζ419+ζ418+ζ415 ζ4126+ζ4119+ζ4117+ζ4114+ζ416 ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 ζ4137+ζ4118+ζ4116+ζ4110+ζ41 ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 complex faithful ρ12 5 0 0 0 0 ζ4139+ζ4121+ζ419+ζ418+ζ415 ζ4126+ζ4119+ζ4117+ζ4114+ζ416 ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 ζ4137+ζ4118+ζ4116+ζ4110+ζ41 ζ4136+ζ4133+ζ4132+ζ4120+ζ412 ζ4130+ζ4129+ζ4113+ζ417+ζ413 ζ4140+ζ4131+ζ4125+ζ4123+ζ414 complex faithful ρ13 5 0 0 0 0 ζ4136+ζ4133+ζ4132+ζ4120+ζ412 ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 ζ4130+ζ4129+ζ4113+ζ417+ζ413 ζ4126+ζ4119+ζ4117+ζ4114+ζ416 ζ4140+ζ4131+ζ4125+ζ4123+ζ414 ζ4139+ζ4121+ζ419+ζ418+ζ415 ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 ζ4137+ζ4118+ζ4116+ζ4110+ζ41 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)

 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 408 787 471 550
,
 1 0 0 0 0 313 753 518 596 433 577 780 503 322 39 239 807 32 619 802 545 752 453 594 587

`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`

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