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## G = Q8.F5order 160 = 25·5

### The non-split extension by Q8 of F5 acting through Inn(Q8)

Aliases: Q8.F5, D20.C4, Dic5.13C23, (C5×Q8).C4, D5⋊C83C2, C52(C8○D4), C4.F54C2, C4.6(C2×F5), C20.6(C2×C4), C5⋊C8.2C22, D10.2(C2×C4), C10.9(C22×C4), Q82D5.3C2, C2.10(C22×F5), (C4×D5).13C22, SmallGroup(160,208)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Q8.F5
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — D5⋊C8 — Q8.F5
 Lower central C5 — C10 — Q8.F5
 Upper central C1 — C2 — Q8

Generators and relations for Q8.F5
G = < a,b,c,d | a4=c5=1, b2=d4=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 180 in 62 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, D4, Q8, D5, C10, C2×C8, M4(2), C4○D4, Dic5, C20, D10, C8○D4, C5⋊C8, C5⋊C8, C4×D5, D20, C5×Q8, D5⋊C8, C4.F5, Q82D5, Q8.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, F5, C8○D4, C2×F5, C22×F5, Q8.F5

Character table of Q8.F5

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10 20A 20B 20C size 1 1 10 10 10 2 2 2 5 5 4 5 5 5 5 10 10 10 10 10 10 4 8 8 8 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 1 1 -1 1 1 -1 -1 1 -1 -1 1 i -i -i i -i -i i i -i i 1 -1 1 -1 linear of order 4 ρ10 1 1 1 1 -1 1 -1 -1 -1 -1 1 i -i -i i i i -i i -i -i 1 1 -1 -1 linear of order 4 ρ11 1 1 -1 -1 -1 1 1 1 -1 -1 1 -i i i -i -i i -i i -i i 1 1 1 1 linear of order 4 ρ12 1 1 1 -1 1 -1 1 -1 -1 -1 1 -i i i -i i -i i i -i -i 1 -1 -1 1 linear of order 4 ρ13 1 1 1 1 -1 1 -1 -1 -1 -1 1 -i i i -i -i -i i -i i i 1 1 -1 -1 linear of order 4 ρ14 1 1 -1 1 1 -1 -1 1 -1 -1 1 -i i i -i i i -i -i i -i 1 -1 1 -1 linear of order 4 ρ15 1 1 1 -1 1 -1 1 -1 -1 -1 1 i -i -i i -i i -i -i i i 1 -1 -1 1 linear of order 4 ρ16 1 1 -1 -1 -1 1 1 1 -1 -1 1 i -i -i i i -i i -i i -i 1 1 1 1 linear of order 4 ρ17 2 -2 0 0 0 0 0 0 2i -2i 2 2ζ87 2ζ8 2ζ85 2ζ83 0 0 0 0 0 0 -2 0 0 0 complex lifted from C8○D4 ρ18 2 -2 0 0 0 0 0 0 -2i 2i 2 2ζ85 2ζ83 2ζ87 2ζ8 0 0 0 0 0 0 -2 0 0 0 complex lifted from C8○D4 ρ19 2 -2 0 0 0 0 0 0 2i -2i 2 2ζ83 2ζ85 2ζ8 2ζ87 0 0 0 0 0 0 -2 0 0 0 complex lifted from C8○D4 ρ20 2 -2 0 0 0 0 0 0 -2i 2i 2 2ζ8 2ζ87 2ζ83 2ζ85 0 0 0 0 0 0 -2 0 0 0 complex lifted from C8○D4 ρ21 4 4 0 0 0 -4 4 -4 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from C2×F5 ρ22 4 4 0 0 0 4 4 4 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ23 4 4 0 0 0 -4 -4 4 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 orthogonal lifted from C2×F5 ρ24 4 4 0 0 0 4 -4 -4 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from C2×F5 ρ25 8 -8 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of Q8.F5
On 80 points
Generators in S80
```(1 29 5 25)(2 30 6 26)(3 31 7 27)(4 32 8 28)(9 63 13 59)(10 64 14 60)(11 57 15 61)(12 58 16 62)(17 41 21 45)(18 42 22 46)(19 43 23 47)(20 44 24 48)(33 51 37 55)(34 52 38 56)(35 53 39 49)(36 54 40 50)(65 77 69 73)(66 78 70 74)(67 79 71 75)(68 80 72 76)
(1 7 5 3)(2 8 6 4)(9 65 13 69)(10 66 14 70)(11 67 15 71)(12 68 16 72)(17 35 21 39)(18 36 22 40)(19 37 23 33)(20 38 24 34)(25 27 29 31)(26 28 30 32)(41 49 45 53)(42 50 46 54)(43 51 47 55)(44 52 48 56)(57 75 61 79)(58 76 62 80)(59 77 63 73)(60 78 64 74)
(1 21 75 63 33)(2 64 22 34 76)(3 35 57 77 23)(4 78 36 24 58)(5 17 79 59 37)(6 60 18 38 80)(7 39 61 73 19)(8 74 40 20 62)(9 55 25 41 71)(10 42 56 72 26)(11 65 43 27 49)(12 28 66 50 44)(13 51 29 45 67)(14 46 52 68 30)(15 69 47 31 53)(16 32 70 54 48)
(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)| (1,29,5,25)(2,30,6,26)(3,31,7,27)(4,32,8,28)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,41,21,45)(18,42,22,46)(19,43,23,47)(20,44,24,48)(33,51,37,55)(34,52,38,56)(35,53,39,49)(36,54,40,50)(65,77,69,73)(66,78,70,74)(67,79,71,75)(68,80,72,76), (1,7,5,3)(2,8,6,4)(9,65,13,69)(10,66,14,70)(11,67,15,71)(12,68,16,72)(17,35,21,39)(18,36,22,40)(19,37,23,33)(20,38,24,34)(25,27,29,31)(26,28,30,32)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56)(57,75,61,79)(58,76,62,80)(59,77,63,73)(60,78,64,74), (1,21,75,63,33)(2,64,22,34,76)(3,35,57,77,23)(4,78,36,24,58)(5,17,79,59,37)(6,60,18,38,80)(7,39,61,73,19)(8,74,40,20,62)(9,55,25,41,71)(10,42,56,72,26)(11,65,43,27,49)(12,28,66,50,44)(13,51,29,45,67)(14,46,52,68,30)(15,69,47,31,53)(16,32,70,54,48), (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( (1,29,5,25)(2,30,6,26)(3,31,7,27)(4,32,8,28)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,41,21,45)(18,42,22,46)(19,43,23,47)(20,44,24,48)(33,51,37,55)(34,52,38,56)(35,53,39,49)(36,54,40,50)(65,77,69,73)(66,78,70,74)(67,79,71,75)(68,80,72,76), (1,7,5,3)(2,8,6,4)(9,65,13,69)(10,66,14,70)(11,67,15,71)(12,68,16,72)(17,35,21,39)(18,36,22,40)(19,37,23,33)(20,38,24,34)(25,27,29,31)(26,28,30,32)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56)(57,75,61,79)(58,76,62,80)(59,77,63,73)(60,78,64,74), (1,21,75,63,33)(2,64,22,34,76)(3,35,57,77,23)(4,78,36,24,58)(5,17,79,59,37)(6,60,18,38,80)(7,39,61,73,19)(8,74,40,20,62)(9,55,25,41,71)(10,42,56,72,26)(11,65,43,27,49)(12,28,66,50,44)(13,51,29,45,67)(14,46,52,68,30)(15,69,47,31,53)(16,32,70,54,48), (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([[(1,29,5,25),(2,30,6,26),(3,31,7,27),(4,32,8,28),(9,63,13,59),(10,64,14,60),(11,57,15,61),(12,58,16,62),(17,41,21,45),(18,42,22,46),(19,43,23,47),(20,44,24,48),(33,51,37,55),(34,52,38,56),(35,53,39,49),(36,54,40,50),(65,77,69,73),(66,78,70,74),(67,79,71,75),(68,80,72,76)], [(1,7,5,3),(2,8,6,4),(9,65,13,69),(10,66,14,70),(11,67,15,71),(12,68,16,72),(17,35,21,39),(18,36,22,40),(19,37,23,33),(20,38,24,34),(25,27,29,31),(26,28,30,32),(41,49,45,53),(42,50,46,54),(43,51,47,55),(44,52,48,56),(57,75,61,79),(58,76,62,80),(59,77,63,73),(60,78,64,74)], [(1,21,75,63,33),(2,64,22,34,76),(3,35,57,77,23),(4,78,36,24,58),(5,17,79,59,37),(6,60,18,38,80),(7,39,61,73,19),(8,74,40,20,62),(9,55,25,41,71),(10,42,56,72,26),(11,65,43,27,49),(12,28,66,50,44),(13,51,29,45,67),(14,46,52,68,30),(15,69,47,31,53),(16,32,70,54,48)], [(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)]])`

Q8.F5 is a maximal subgroup of
SD163F5  SD162F5  Q165F5  Q16⋊F5  Dic5.20C24  Dic5.21C24  Dic5.22C24  SL2(𝔽3).F5  D60.C4  Dic6.F5  D20.Dic3
Q8.F5 is a maximal quotient of
D10.C42  D202C8  D102M4(2)  C20⋊M4(2)  C4⋊C4.7F5  C4⋊C4.9F5  Q8×C5⋊C8  (C2×Q8).5F5  C20.6M4(2)  D60.C4  Dic6.F5  D20.Dic3

Matrix representation of Q8.F5 in GL6(𝔽41)

 40 39 0 0 0 0 1 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 32 0 0 0 0 0 9 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3 38 19 0 0 0 22 38 0 3 0 0 3 0 38 22 0 0 0 19 38 3

`G:=sub<GL(6,GF(41))| [40,1,0,0,0,0,39,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,9,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,22,3,0,0,0,38,38,0,19,0,0,19,0,38,38,0,0,0,3,22,3] >;`

Q8.F5 in GAP, Magma, Sage, TeX

`Q_8.F_5`
`% in TeX`

`G:=Group("Q8.F5");`
`// GroupNames label`

`G:=SmallGroup(160,208);`
`// by ID`

`G=gap.SmallGroup(160,208);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,188,86,69,2309,599]);`
`// Polycyclic`

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

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