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## G = C8.F5order 160 = 25·5

### 3rd non-split extension by C8 of F5 acting via F5/D5=C2

Aliases: C8.3F5, C40.5C4, C51M5(2), D10.3C8, Dic5.3C8, C5⋊C161C2, C10.2(C2×C8), (C4×D5).6C4, (C8×D5).8C2, C4.15(C2×F5), C20.14(C2×C4), C2.3(D5⋊C8), C52C8.15C22, SmallGroup(160,65)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C8.F5
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C8.F5
 Lower central C5 — C10 — C8.F5
 Upper central C1 — C4 — C8

Generators and relations for C8.F5
G = < a,b,c | a8=b5=1, c4=a2, ab=ba, cac-1=a5, cbc-1=b3 >

Character table of C8.F5

 class 1 2A 2B 4A 4B 4C 5 8A 8B 8C 8D 8E 8F 10 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 40A 40B 40C 40D size 1 1 10 1 1 10 4 2 2 5 5 5 5 4 10 10 10 10 10 10 10 10 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 1 -1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 i i -i -i -i -i i i 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -i -i i i i i -i -i 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 1 i -i i i -i -i i -i 1 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -i i -i -i i i -i i 1 1 1 1 1 1 linear of order 4 ρ9 1 1 -1 -1 -1 1 1 -i i i -i i -i 1 ζ87 ζ87 ζ8 ζ85 ζ85 ζ8 ζ83 ζ83 -1 -1 i -i i -i linear of order 8 ρ10 1 1 -1 -1 -1 1 1 -i i i -i i -i 1 ζ83 ζ83 ζ85 ζ8 ζ8 ζ85 ζ87 ζ87 -1 -1 i -i i -i linear of order 8 ρ11 1 1 1 -1 -1 -1 1 -i i -i i -i i 1 ζ85 ζ8 ζ87 ζ83 ζ87 ζ83 ζ8 ζ85 -1 -1 i -i i -i linear of order 8 ρ12 1 1 1 -1 -1 -1 1 -i i -i i -i i 1 ζ8 ζ85 ζ83 ζ87 ζ83 ζ87 ζ85 ζ8 -1 -1 i -i i -i linear of order 8 ρ13 1 1 -1 -1 -1 1 1 i -i -i i -i i 1 ζ85 ζ85 ζ83 ζ87 ζ87 ζ83 ζ8 ζ8 -1 -1 -i i -i i linear of order 8 ρ14 1 1 1 -1 -1 -1 1 i -i i -i i -i 1 ζ83 ζ87 ζ8 ζ85 ζ8 ζ85 ζ87 ζ83 -1 -1 -i i -i i linear of order 8 ρ15 1 1 1 -1 -1 -1 1 i -i i -i i -i 1 ζ87 ζ83 ζ85 ζ8 ζ85 ζ8 ζ83 ζ87 -1 -1 -i i -i i linear of order 8 ρ16 1 1 -1 -1 -1 1 1 i -i -i i -i i 1 ζ8 ζ8 ζ87 ζ83 ζ83 ζ87 ζ85 ζ85 -1 -1 -i i -i i linear of order 8 ρ17 2 -2 0 -2i 2i 0 2 0 0 2ζ8 2ζ87 2ζ85 2ζ83 -2 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 complex lifted from M5(2) ρ18 2 -2 0 2i -2i 0 2 0 0 2ζ87 2ζ8 2ζ83 2ζ85 -2 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 complex lifted from M5(2) ρ19 2 -2 0 2i -2i 0 2 0 0 2ζ83 2ζ85 2ζ87 2ζ8 -2 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 complex lifted from M5(2) ρ20 2 -2 0 -2i 2i 0 2 0 0 2ζ85 2ζ83 2ζ8 2ζ87 -2 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 complex lifted from M5(2) ρ21 4 4 0 4 4 0 -1 4 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ22 4 4 0 4 4 0 -1 -4 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ23 4 4 0 -4 -4 0 -1 4i -4i 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 i -i i -i complex lifted from D5⋊C8, Schur index 2 ρ24 4 4 0 -4 -4 0 -1 -4i 4i 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -i i -i i complex lifted from D5⋊C8, Schur index 2 ρ25 4 -4 0 4i -4i 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 i -i 2ζ8ζ54+2ζ8ζ5+ζ8 2ζ83ζ54+2ζ83ζ5+ζ83 2ζ85ζ54+2ζ85ζ5+ζ85 2ζ83ζ53+2ζ83ζ52+ζ83 complex faithful, Schur index 2 ρ26 4 -4 0 -4i 4i 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -i i 2ζ83ζ53+2ζ83ζ52+ζ83 2ζ85ζ54+2ζ85ζ5+ζ85 2ζ83ζ54+2ζ83ζ5+ζ83 2ζ8ζ54+2ζ8ζ5+ζ8 complex faithful, Schur index 2 ρ27 4 -4 0 4i -4i 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 i -i 2ζ85ζ54+2ζ85ζ5+ζ85 2ζ83ζ53+2ζ83ζ52+ζ83 2ζ8ζ54+2ζ8ζ5+ζ8 2ζ83ζ54+2ζ83ζ5+ζ83 complex faithful, Schur index 2 ρ28 4 -4 0 -4i 4i 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -i i 2ζ83ζ54+2ζ83ζ5+ζ83 2ζ8ζ54+2ζ8ζ5+ζ8 2ζ83ζ53+2ζ83ζ52+ζ83 2ζ85ζ54+2ζ85ζ5+ζ85 complex faithful, Schur index 2

Smallest permutation representation of C8.F5
On 80 points
Generators in S80
```(1 11 5 15 9 3 13 7)(2 4 6 8 10 12 14 16)(17 62 21 50 25 54 29 58)(18 55 22 59 26 63 30 51)(19 64 23 52 27 56 31 60)(20 57 24 61 28 49 32 53)(33 70 37 74 41 78 45 66)(34 79 38 67 42 71 46 75)(35 72 39 76 43 80 47 68)(36 65 40 69 44 73 48 77)
(1 36 23 58 71)(2 59 37 72 24)(3 73 60 25 38)(4 26 74 39 61)(5 40 27 62 75)(6 63 41 76 28)(7 77 64 29 42)(8 30 78 43 49)(9 44 31 50 79)(10 51 45 80 32)(11 65 52 17 46)(12 18 66 47 53)(13 48 19 54 67)(14 55 33 68 20)(15 69 56 21 34)(16 22 70 35 57)
(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 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)```

`G:=sub<Sym(80)| (1,11,5,15,9,3,13,7)(2,4,6,8,10,12,14,16)(17,62,21,50,25,54,29,58)(18,55,22,59,26,63,30,51)(19,64,23,52,27,56,31,60)(20,57,24,61,28,49,32,53)(33,70,37,74,41,78,45,66)(34,79,38,67,42,71,46,75)(35,72,39,76,43,80,47,68)(36,65,40,69,44,73,48,77), (1,36,23,58,71)(2,59,37,72,24)(3,73,60,25,38)(4,26,74,39,61)(5,40,27,62,75)(6,63,41,76,28)(7,77,64,29,42)(8,30,78,43,49)(9,44,31,50,79)(10,51,45,80,32)(11,65,52,17,46)(12,18,66,47,53)(13,48,19,54,67)(14,55,33,68,20)(15,69,56,21,34)(16,22,70,35,57), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)>;`

`G:=Group( (1,11,5,15,9,3,13,7)(2,4,6,8,10,12,14,16)(17,62,21,50,25,54,29,58)(18,55,22,59,26,63,30,51)(19,64,23,52,27,56,31,60)(20,57,24,61,28,49,32,53)(33,70,37,74,41,78,45,66)(34,79,38,67,42,71,46,75)(35,72,39,76,43,80,47,68)(36,65,40,69,44,73,48,77), (1,36,23,58,71)(2,59,37,72,24)(3,73,60,25,38)(4,26,74,39,61)(5,40,27,62,75)(6,63,41,76,28)(7,77,64,29,42)(8,30,78,43,49)(9,44,31,50,79)(10,51,45,80,32)(11,65,52,17,46)(12,18,66,47,53)(13,48,19,54,67)(14,55,33,68,20)(15,69,56,21,34)(16,22,70,35,57), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80) );`

`G=PermutationGroup([[(1,11,5,15,9,3,13,7),(2,4,6,8,10,12,14,16),(17,62,21,50,25,54,29,58),(18,55,22,59,26,63,30,51),(19,64,23,52,27,56,31,60),(20,57,24,61,28,49,32,53),(33,70,37,74,41,78,45,66),(34,79,38,67,42,71,46,75),(35,72,39,76,43,80,47,68),(36,65,40,69,44,73,48,77)], [(1,36,23,58,71),(2,59,37,72,24),(3,73,60,25,38),(4,26,74,39,61),(5,40,27,62,75),(6,63,41,76,28),(7,77,64,29,42),(8,30,78,43,49),(9,44,31,50,79),(10,51,45,80,32),(11,65,52,17,46),(12,18,66,47,53),(13,48,19,54,67),(14,55,33,68,20),(15,69,56,21,34),(16,22,70,35,57)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)]])`

C8.F5 is a maximal subgroup of
C16⋊F5  C164F5  D10.D8  D40.C4  D401C4  Dic20.C4  D5⋊M5(2)  Dic10.C8  D30.C8  C120.C4
C8.F5 is a maximal quotient of
C40.C8  D10⋊C16  C10.M5(2)  D30.C8  C120.C4

Matrix representation of C8.F5 in GL6(𝔽241)

 211 0 0 0 0 0 0 30 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 240 0 0 1 0 0 240 0 0 0 1 0 240 0 0 0 0 1 240
,
 0 1 0 0 0 0 30 0 0 0 0 0 0 0 182 59 33 0 0 0 215 59 0 182 0 0 182 0 59 215 0 0 0 33 59 182

`G:=sub<GL(6,GF(241))| [211,0,0,0,0,0,0,30,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[0,30,0,0,0,0,1,0,0,0,0,0,0,0,182,215,182,0,0,0,59,59,0,33,0,0,33,0,59,59,0,0,0,182,215,182] >;`

C8.F5 in GAP, Magma, Sage, TeX

`C_8.F_5`
`% in TeX`

`G:=Group("C8.F5");`
`// GroupNames label`

`G:=SmallGroup(160,65);`
`// by ID`

`G=gap.SmallGroup(160,65);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,50,69,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^5=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^3>;`
`// generators/relations`

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