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G = C41⋊C4order 164 = 22·41

The semidirect product of C41 and C4 acting faithfully

Aliases: C41⋊C4, D41.C2, SmallGroup(164,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C41 — C41⋊C4
 Chief series C1 — C41 — D41 — C41⋊C4
 Lower central C41 — C41⋊C4
 Upper central C1

Generators and relations for C41⋊C4
G = < a,b | a41=b4=1, bab-1=a9 >

Character table of C41⋊C4

 class 1 2 4A 4B 41A 41B 41C 41D 41E 41F 41G 41H 41I 41J size 1 41 41 41 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 -i i 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 i -i 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ5 4 0 0 0 ζ4139+ζ4123+ζ4118+ζ412 ζ4138+ζ4127+ζ4114+ζ413 ζ4130+ζ4124+ζ4117+ζ4111 ζ4137+ζ4136+ζ415+ζ414 ζ4129+ζ4126+ζ4115+ζ4112 ζ4135+ζ4128+ζ4113+ζ416 ζ4133+ζ4131+ζ4110+ζ418 ζ4125+ζ4121+ζ4120+ζ4116 ζ4140+ζ4132+ζ419+ζ41 ζ4134+ζ4122+ζ4119+ζ417 orthogonal faithful ρ6 4 0 0 0 ζ4129+ζ4126+ζ4115+ζ4112 ζ4139+ζ4123+ζ4118+ζ412 ζ4125+ζ4121+ζ4120+ζ4116 ζ4130+ζ4124+ζ4117+ζ4111 ζ4133+ζ4131+ζ4110+ζ418 ζ4137+ζ4136+ζ415+ζ414 ζ4134+ζ4122+ζ4119+ζ417 ζ4138+ζ4127+ζ4114+ζ413 ζ4135+ζ4128+ζ4113+ζ416 ζ4140+ζ4132+ζ419+ζ41 orthogonal faithful ρ7 4 0 0 0 ζ4130+ζ4124+ζ4117+ζ4111 ζ4137+ζ4136+ζ415+ζ414 ζ4140+ζ4132+ζ419+ζ41 ζ4134+ζ4122+ζ4119+ζ417 ζ4125+ζ4121+ζ4120+ζ4116 ζ4133+ζ4131+ζ4110+ζ418 ζ4138+ζ4127+ζ4114+ζ413 ζ4135+ζ4128+ζ4113+ζ416 ζ4129+ζ4126+ζ4115+ζ4112 ζ4139+ζ4123+ζ4118+ζ412 orthogonal faithful ρ8 4 0 0 0 ζ4138+ζ4127+ζ4114+ζ413 ζ4125+ζ4121+ζ4120+ζ4116 ζ4137+ζ4136+ζ415+ζ414 ζ4135+ζ4128+ζ4113+ζ416 ζ4139+ζ4123+ζ4118+ζ412 ζ4140+ζ4132+ζ419+ζ41 ζ4129+ζ4126+ζ4115+ζ4112 ζ4130+ζ4124+ζ4117+ζ4111 ζ4134+ζ4122+ζ4119+ζ417 ζ4133+ζ4131+ζ4110+ζ418 orthogonal faithful ρ9 4 0 0 0 ζ4140+ζ4132+ζ419+ζ41 ζ4134+ζ4122+ζ4119+ζ417 ζ4129+ζ4126+ζ4115+ζ4112 ζ4139+ζ4123+ζ4118+ζ412 ζ4135+ζ4128+ζ4113+ζ416 ζ4138+ζ4127+ζ4114+ζ413 ζ4137+ζ4136+ζ415+ζ414 ζ4133+ζ4131+ζ4110+ζ418 ζ4125+ζ4121+ζ4120+ζ4116 ζ4130+ζ4124+ζ4117+ζ4111 orthogonal faithful ρ10 4 0 0 0 ζ4135+ζ4128+ζ4113+ζ416 ζ4140+ζ4132+ζ419+ζ41 ζ4133+ζ4131+ζ4110+ζ418 ζ4129+ζ4126+ζ4115+ζ4112 ζ4137+ζ4136+ζ415+ζ414 ζ4139+ζ4123+ζ4118+ζ412 ζ4130+ζ4124+ζ4117+ζ4111 ζ4134+ζ4122+ζ4119+ζ417 ζ4138+ζ4127+ζ4114+ζ413 ζ4125+ζ4121+ζ4120+ζ4116 orthogonal faithful ρ11 4 0 0 0 ζ4137+ζ4136+ζ415+ζ414 ζ4135+ζ4128+ζ4113+ζ416 ζ4134+ζ4122+ζ4119+ζ417 ζ4133+ζ4131+ζ4110+ζ418 ζ4130+ζ4124+ζ4117+ζ4111 ζ4129+ζ4126+ζ4115+ζ4112 ζ4125+ζ4121+ζ4120+ζ4116 ζ4140+ζ4132+ζ419+ζ41 ζ4139+ζ4123+ζ4118+ζ412 ζ4138+ζ4127+ζ4114+ζ413 orthogonal faithful ρ12 4 0 0 0 ζ4125+ζ4121+ζ4120+ζ4116 ζ4130+ζ4124+ζ4117+ζ4111 ζ4135+ζ4128+ζ4113+ζ416 ζ4140+ζ4132+ζ419+ζ41 ζ4138+ζ4127+ζ4114+ζ413 ζ4134+ζ4122+ζ4119+ζ417 ζ4139+ζ4123+ζ4118+ζ412 ζ4137+ζ4136+ζ415+ζ414 ζ4133+ζ4131+ζ4110+ζ418 ζ4129+ζ4126+ζ4115+ζ4112 orthogonal faithful ρ13 4 0 0 0 ζ4133+ζ4131+ζ4110+ζ418 ζ4129+ζ4126+ζ4115+ζ4112 ζ4138+ζ4127+ζ4114+ζ413 ζ4125+ζ4121+ζ4120+ζ4116 ζ4134+ζ4122+ζ4119+ζ417 ζ4130+ζ4124+ζ4117+ζ4111 ζ4140+ζ4132+ζ419+ζ41 ζ4139+ζ4123+ζ4118+ζ412 ζ4137+ζ4136+ζ415+ζ414 ζ4135+ζ4128+ζ4113+ζ416 orthogonal faithful ρ14 4 0 0 0 ζ4134+ζ4122+ζ4119+ζ417 ζ4133+ζ4131+ζ4110+ζ418 ζ4139+ζ4123+ζ4118+ζ412 ζ4138+ζ4127+ζ4114+ζ413 ζ4140+ζ4132+ζ419+ζ41 ζ4125+ζ4121+ζ4120+ζ4116 ζ4135+ζ4128+ζ4113+ζ416 ζ4129+ζ4126+ζ4115+ζ4112 ζ4130+ζ4124+ζ4117+ζ4111 ζ4137+ζ4136+ζ415+ζ414 orthogonal faithful

Smallest permutation representation of C41⋊C4
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 33 41 10)(3 24 40 19)(4 15 39 28)(5 6 38 37)(7 29 36 14)(8 20 35 23)(9 11 34 32)(12 25 31 18)(13 16 30 27)(17 21 26 22)```

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

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

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

C41⋊C4 is a maximal subgroup of   C41⋊C8  C41⋊Dic3
C41⋊C4 is a maximal quotient of   C412C8  C41⋊Dic3

Matrix representation of C41⋊C4 in GL4(𝔽821) generated by

 817 1 0 0 267 0 1 0 554 0 0 1 109 756 401 789
,
 787 474 393 437 47 277 249 307 652 653 448 756 446 396 609 130
`G:=sub<GL(4,GF(821))| [817,267,554,109,1,0,0,756,0,1,0,401,0,0,1,789],[787,47,652,446,474,277,653,396,393,249,448,609,437,307,756,130] >;`

C41⋊C4 in GAP, Magma, Sage, TeX

`C_{41}\rtimes C_4`
`% in TeX`

`G:=Group("C41:C4");`
`// GroupNames label`

`G:=SmallGroup(164,3);`
`// by ID`

`G=gap.SmallGroup(164,3);`
`# by ID`

`G:=PCGroup([3,-2,-2,-41,6,1154,725]);`
`// Polycyclic`

`G:=Group<a,b|a^41=b^4=1,b*a*b^-1=a^9>;`
`// generators/relations`

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