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## G = C41⋊C8order 328 = 23·41

### The semidirect product of C41 and C8 acting faithfully

Aliases: C41⋊C8, D41.C4, C41⋊C4.C2, SmallGroup(328,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C41 — C41⋊C8
 Chief series C1 — C41 — D41 — C41⋊C4 — C41⋊C8
 Lower central C41 — C41⋊C8
 Upper central C1

Generators and relations for C41⋊C8
G = < a,b | a41=b8=1, bab-1=a38 >

Character table of C41⋊C8

 class 1 2 4A 4B 8A 8B 8C 8D 41A 41B 41C 41D 41E size 1 41 41 41 41 41 41 41 8 8 8 8 8 ρ1 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 linear of order 2 ρ3 1 1 -1 -1 i -i -i i 1 1 1 1 1 linear of order 4 ρ4 1 1 -1 -1 -i i i -i 1 1 1 1 1 linear of order 4 ρ5 1 -1 i -i ζ85 ζ83 ζ87 ζ8 1 1 1 1 1 linear of order 8 ρ6 1 -1 -i i ζ87 ζ8 ζ85 ζ83 1 1 1 1 1 linear of order 8 ρ7 1 -1 -i i ζ83 ζ85 ζ8 ζ87 1 1 1 1 1 linear of order 8 ρ8 1 -1 i -i ζ8 ζ87 ζ83 ζ85 1 1 1 1 1 linear of order 8 ρ9 8 0 0 0 0 0 0 0 ζ4134+ζ4125+ζ4122+ζ4121+ζ4120+ζ4119+ζ4116+ζ417 ζ4133+ζ4131+ζ4130+ζ4124+ζ4117+ζ4111+ζ4110+ζ418 ζ4140+ζ4138+ζ4132+ζ4127+ζ4114+ζ419+ζ413+ζ41 ζ4137+ζ4136+ζ4129+ζ4126+ζ4115+ζ4112+ζ415+ζ414 ζ4139+ζ4135+ζ4128+ζ4123+ζ4118+ζ4113+ζ416+ζ412 orthogonal faithful ρ10 8 0 0 0 0 0 0 0 ζ4139+ζ4135+ζ4128+ζ4123+ζ4118+ζ4113+ζ416+ζ412 ζ4140+ζ4138+ζ4132+ζ4127+ζ4114+ζ419+ζ413+ζ41 ζ4137+ζ4136+ζ4129+ζ4126+ζ4115+ζ4112+ζ415+ζ414 ζ4134+ζ4125+ζ4122+ζ4121+ζ4120+ζ4119+ζ4116+ζ417 ζ4133+ζ4131+ζ4130+ζ4124+ζ4117+ζ4111+ζ4110+ζ418 orthogonal faithful ρ11 8 0 0 0 0 0 0 0 ζ4140+ζ4138+ζ4132+ζ4127+ζ4114+ζ419+ζ413+ζ41 ζ4134+ζ4125+ζ4122+ζ4121+ζ4120+ζ4119+ζ4116+ζ417 ζ4139+ζ4135+ζ4128+ζ4123+ζ4118+ζ4113+ζ416+ζ412 ζ4133+ζ4131+ζ4130+ζ4124+ζ4117+ζ4111+ζ4110+ζ418 ζ4137+ζ4136+ζ4129+ζ4126+ζ4115+ζ4112+ζ415+ζ414 orthogonal faithful ρ12 8 0 0 0 0 0 0 0 ζ4137+ζ4136+ζ4129+ζ4126+ζ4115+ζ4112+ζ415+ζ414 ζ4139+ζ4135+ζ4128+ζ4123+ζ4118+ζ4113+ζ416+ζ412 ζ4133+ζ4131+ζ4130+ζ4124+ζ4117+ζ4111+ζ4110+ζ418 ζ4140+ζ4138+ζ4132+ζ4127+ζ4114+ζ419+ζ413+ζ41 ζ4134+ζ4125+ζ4122+ζ4121+ζ4120+ζ4119+ζ4116+ζ417 orthogonal faithful ρ13 8 0 0 0 0 0 0 0 ζ4133+ζ4131+ζ4130+ζ4124+ζ4117+ζ4111+ζ4110+ζ418 ζ4137+ζ4136+ζ4129+ζ4126+ζ4115+ζ4112+ζ415+ζ414 ζ4134+ζ4125+ζ4122+ζ4121+ζ4120+ζ4119+ζ4116+ζ417 ζ4139+ζ4135+ζ4128+ζ4123+ζ4118+ζ4113+ζ416+ζ412 ζ4140+ζ4138+ζ4132+ζ4127+ζ4114+ζ419+ζ413+ζ41 orthogonal faithful

Smallest permutation representation of C41⋊C8
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 28 33 4 41 15 10 39)(3 14 24 7 40 29 19 36)(5 27 6 13 38 16 37 30)(8 26 20 22 35 17 23 21)(9 12 11 25 34 31 32 18)```

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

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

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

Matrix representation of C41⋊C8 in GL8(𝔽2297)

 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2296 2270 1712 2242 1711 2242 1712 2270
,
 1 0 0 0 0 0 0 0 367 1648 956 1505 957 1477 733 2153 847 72 1126 95 1520 878 427 2177 1488 2188 270 236 191 261 2002 591 1927 2258 658 1693 70 1831 273 730 1014 1717 1805 288 295 1806 1915 1035 1283 1221 1351 1164 1014 1943 1241 1903 370 1532 805 1508 972 1370 1190 763

`G:=sub<GL(8,GF(2297))| [0,0,0,0,0,0,0,2296,1,0,0,0,0,0,0,2270,0,1,0,0,0,0,0,1712,0,0,1,0,0,0,0,2242,0,0,0,1,0,0,0,1711,0,0,0,0,1,0,0,2242,0,0,0,0,0,1,0,1712,0,0,0,0,0,0,1,2270],[1,367,847,1488,1927,1014,1283,370,0,1648,72,2188,2258,1717,1221,1532,0,956,1126,270,658,1805,1351,805,0,1505,95,236,1693,288,1164,1508,0,957,1520,191,70,295,1014,972,0,1477,878,261,1831,1806,1943,1370,0,733,427,2002,273,1915,1241,1190,0,2153,2177,591,730,1035,1903,763] >;`

C41⋊C8 in GAP, Magma, Sage, TeX

`C_{41}\rtimes C_8`
`% in TeX`

`G:=Group("C41:C8");`
`// GroupNames label`

`G:=SmallGroup(328,12);`
`// by ID`

`G=gap.SmallGroup(328,12);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-41,8,21,3459,2055,1291]);`
`// Polycyclic`

`G:=Group<a,b|a^41=b^8=1,b*a*b^-1=a^38>;`
`// generators/relations`

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