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## G = D8⋊F5order 320 = 26·5

### 4th semidirect product of D8 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D8⋊F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — D5⋊C8 — D4.F5 — D8⋊F5
 Lower central C5 — C10 — C20 — D8⋊F5
 Upper central C1 — C2 — C4 — D8

Generators and relations for D8⋊F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 386 in 104 conjugacy classes, 40 normal (22 characteristic)
C1, C2, C2 [×3], C4, C4 [×4], C22 [×3], C5, C8, C8 [×5], C2×C4 [×4], D4 [×2], D4 [×2], Q8 [×2], D5, C10, C10 [×2], C42, C2×C8 [×4], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, Dic5 [×2], C20, F5, D10, C2×C10 [×2], C8⋊C4, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8, C40, C5⋊C8 [×2], C5⋊C8 [×2], Dic10 [×2], C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4 [×2], C2×F5, C8.26D4, C8×D5, Dic20, D4.D5 [×2], C5×D8, D5⋊C8, C4.F5 [×2], C4×F5, C2×C5⋊C8 [×2], C22.F5 [×2], D42D5 [×2], C8⋊F5, C40.C4, D4⋊F5 [×2], D83D5, D4.F5 [×2], D8⋊F5
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], C8.26D4, C22×F5, D4×F5, D8⋊F5

Character table of D8⋊F5

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 5 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 10C 20 40A 40B size 1 1 4 4 10 2 5 5 20 20 20 20 4 4 10 10 10 10 20 20 20 20 20 4 16 16 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 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 -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 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 -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 -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 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 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 -1 1 1 1 linear of order 2 ρ9 1 1 1 1 -1 1 -1 -1 i -1 -i -1 1 1 -i i i -i -1 -i i -i i 1 1 1 1 1 1 linear of order 4 ρ10 1 1 1 -1 -1 1 -1 -1 i -1 -i 1 1 -1 i -i -i i 1 -i i i -i 1 1 -1 1 -1 -1 linear of order 4 ρ11 1 1 1 1 -1 1 -1 -1 -i -1 i -1 1 1 i -i -i i -1 i -i i -i 1 1 1 1 1 1 linear of order 4 ρ12 1 1 1 -1 -1 1 -1 -1 -i -1 i 1 1 -1 -i i i -i 1 i -i -i i 1 1 -1 1 -1 -1 linear of order 4 ρ13 1 1 -1 1 -1 1 -1 -1 -i 1 i -1 1 -1 -i i i -i 1 -i i i -i 1 -1 1 1 -1 -1 linear of order 4 ρ14 1 1 -1 -1 -1 1 -1 -1 -i 1 i 1 1 1 i -i -i i -1 -i i -i i 1 -1 -1 1 1 1 linear of order 4 ρ15 1 1 -1 1 -1 1 -1 -1 i 1 -i -1 1 -1 i -i -i i 1 i -i -i i 1 -1 1 1 -1 -1 linear of order 4 ρ16 1 1 -1 -1 -1 1 -1 -1 i 1 -i 1 1 1 -i i i -i -1 i -i i -i 1 -1 -1 1 1 1 linear of order 4 ρ17 2 2 0 0 -2 -2 2 2 0 0 0 0 2 0 -2 2 -2 2 0 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ18 2 2 0 0 -2 -2 2 2 0 0 0 0 2 0 2 -2 2 -2 0 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ19 2 2 0 0 2 -2 -2 -2 0 0 0 0 2 0 2i 2i -2i -2i 0 0 0 0 0 2 0 0 -2 0 0 complex lifted from C4○D4 ρ20 2 2 0 0 2 -2 -2 -2 0 0 0 0 2 0 -2i -2i 2i 2i 0 0 0 0 0 2 0 0 -2 0 0 complex lifted from C4○D4 ρ21 4 4 -4 -4 0 4 0 0 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 0 -1 1 1 -1 -1 -1 orthogonal lifted from C2×F5 ρ22 4 4 -4 4 0 4 0 0 0 0 0 0 -1 -4 0 0 0 0 0 0 0 0 0 -1 1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ23 4 4 4 -4 0 4 0 0 0 0 0 0 -1 -4 0 0 0 0 0 0 0 0 0 -1 -1 1 -1 1 1 orthogonal lifted from C2×F5 ρ24 4 4 4 4 0 4 0 0 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ25 4 -4 0 0 0 0 4i -4i 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 complex lifted from C8.26D4 ρ26 4 -4 0 0 0 0 -4i 4i 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 complex lifted from C8.26D4 ρ27 8 8 0 0 0 -8 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 -2 0 0 2 0 0 orthogonal lifted from D4×F5 ρ28 8 -8 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 -√10 √10 symplectic faithful, Schur index 2 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 √10 -√10 symplectic faithful, Schur index 2

Smallest permutation representation of D8⋊F5
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 59)(2 58)(3 57)(4 64)(5 63)(6 62)(7 61)(8 60)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)(41 73)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 72)(56 71)
(1 10 29 77 66)(2 11 30 78 67)(3 12 31 79 68)(4 13 32 80 69)(5 14 25 73 70)(6 15 26 74 71)(7 16 27 75 72)(8 9 28 76 65)(17 46 54 60 38)(18 47 55 61 39)(19 48 56 62 40)(20 41 49 63 33)(21 42 50 64 34)(22 43 51 57 35)(23 44 52 58 36)(24 45 53 59 37)
(2 6)(4 8)(9 32 65 80)(10 29 66 77)(11 26 67 74)(12 31 68 79)(13 28 69 76)(14 25 70 73)(15 30 71 78)(16 27 72 75)(17 52 42 40)(18 49 43 37)(19 54 44 34)(20 51 45 39)(21 56 46 36)(22 53 47 33)(23 50 48 38)(24 55 41 35)(57 59 61 63)(58 64 62 60)```

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

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

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

Matrix representation of D8⋊F5 in GL8(𝔽41)

 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 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 0 9 0 0 0
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 40 40 40 40 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 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
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 40

`G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40] >;`

D8⋊F5 in GAP, Magma, Sage, TeX

`D_8\rtimes F_5`
`% in TeX`

`G:=Group("D8:F5");`
`// GroupNames label`

`G:=SmallGroup(320,1071);`
`// by ID`

`G=gap.SmallGroup(320,1071);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,136,851,438,102,6278,1595]);`
`// Polycyclic`

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

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