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## G = Q16⋊5F5order 320 = 26·5

### The semidirect product of Q16 and F5 acting through Inn(Q16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Q16⋊5F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — D5⋊C8 — Q8.F5 — Q16⋊5F5
 Lower central C5 — C10 — C20 — Q16⋊5F5
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q165F5
G = < a,b,c,d | a8=c5=d4=1, b2=a4, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 418 in 106 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2 [×3], C4, C4 [×5], C22 [×3], C5, C8, C8 [×5], C2×C4 [×4], D4 [×4], Q8 [×2], D5 [×3], C10, C42, C2×C8 [×4], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], F5 [×2], D10, D10 [×2], C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8, C40, C5⋊C8 [×2], C5⋊C8 [×2], C4×D5, C4×D5 [×2], D20 [×2], D20 [×2], C5×Q8 [×2], C2×F5, C8○D8, C8×D5, D40, Q8⋊D5 [×2], C5×Q16, D5⋊C8, D5⋊C8 [×2], C4.F5 [×2], C4.F5 [×2], C4×F5, Q82D5 [×2], C8×F5, D10.Q8, Q82F5 [×2], Q8.D10, Q8.F5 [×2], Q165F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5 [×3], C8○D8, C22×F5, D4×F5, Q165F5

Smallest permutation representation of Q165F5
On 80 points
Generators in S80
```(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)
(1 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 30 13 26)(10 29 14 25)(11 28 15 32)(12 27 16 31)(17 33 21 37)(18 40 22 36)(19 39 23 35)(20 38 24 34)(41 74 45 78)(42 73 46 77)(43 80 47 76)(44 79 48 75)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
(1 10 37 77 66)(2 11 38 78 67)(3 12 39 79 68)(4 13 40 80 69)(5 14 33 73 70)(6 15 34 74 71)(7 16 35 75 72)(8 9 36 76 65)(17 42 50 60 29)(18 43 51 61 30)(19 44 52 62 31)(20 45 53 63 32)(21 46 54 64 25)(22 47 55 57 26)(23 48 56 58 27)(24 41 49 59 28)
(9 36 65 76)(10 37 66 77)(11 38 67 78)(12 39 68 79)(13 40 69 80)(14 33 70 73)(15 34 71 74)(16 35 72 75)(17 52 46 27)(18 53 47 28)(19 54 48 29)(20 55 41 30)(21 56 42 31)(22 49 43 32)(23 50 44 25)(24 51 45 26)(57 59 61 63)(58 60 62 64)```

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 5 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N 10 20A 20B 20C 40A 40B order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 5 8 8 8 8 8 8 8 8 8 8 8 8 8 8 10 20 20 20 40 40 size 1 1 10 20 20 2 4 4 5 5 10 10 10 10 4 2 2 5 5 5 5 10 10 10 10 20 20 20 20 4 8 16 16 8 8

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 8 8 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 C4○D4 C8○D8 F5 C2×F5 C2×F5 D4×F5 Q16⋊5F5 kernel Q16⋊5F5 C8×F5 D10.Q8 Q8⋊2F5 Q8.D10 Q8.F5 D40 Q8⋊D5 C5×Q16 C5⋊C8 D10 C5 Q16 C8 Q8 C2 C1 # reps 1 1 1 2 1 2 2 4 2 2 2 8 1 1 2 1 2

Matrix representation of Q165F5 in GL6(𝔽41)

 27 0 0 0 0 0 26 38 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 4 9 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 40 40 40 40

`G:=sub<GL(6,GF(41))| [27,26,0,0,0,0,0,38,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[1,4,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;`

Q165F5 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes_5F_5`
`% in TeX`

`G:=Group("Q16:5F5");`
`// GroupNames label`

`G:=SmallGroup(320,1078);`
`// by ID`

`G=gap.SmallGroup(320,1078);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,219,184,136,851,438,102,6278,1595]);`
`// Polycyclic`

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

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