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## G = D5⋊M5(2)  order 320 = 26·5

### The semidirect product of D5 and M5(2) acting via M5(2)/C2×C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D5⋊M5(2)
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C20.C8 — D5⋊M5(2)
 Lower central C5 — C10 — D5⋊M5(2)
 Upper central C1 — C8 — C2×C8

Generators and relations for D5⋊M5(2)
G = < a,b,c,d | a5=b2=c16=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd=c9 >

Subgroups: 250 in 90 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, D10, C2×C10, C2×C16, M5(2), C22×C8, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×M5(2), C5⋊C16, C8×D5, C2×C52C8, C2×C40, C2×C4×D5, D5⋊C16, C8.F5, C20.C8, D5×C2×C8, D5⋊M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, F5, M5(2), C22×C8, C2×F5, C2×M5(2), D5⋊C8, C22×F5, C2×D5⋊C8, D5⋊M5(2)

Smallest permutation representation of D5⋊M5(2)
On 80 points
Generators in S80
```(1 23 53 34 66)(2 35 24 67 54)(3 68 36 55 25)(4 56 69 26 37)(5 27 57 38 70)(6 39 28 71 58)(7 72 40 59 29)(8 60 73 30 41)(9 31 61 42 74)(10 43 32 75 62)(11 76 44 63 17)(12 64 77 18 45)(13 19 49 46 78)(14 47 20 79 50)(15 80 48 51 21)(16 52 65 22 33)
(1 66)(2 54)(3 25)(4 37)(5 70)(6 58)(7 29)(8 41)(9 74)(10 62)(11 17)(12 45)(13 78)(14 50)(15 21)(16 33)(18 64)(19 46)(22 52)(23 34)(26 56)(27 38)(30 60)(31 42)(35 67)(39 71)(43 75)(47 79)(51 80)(55 68)(59 72)(63 76)
(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 M5(2) F5 C2×F5 C2×F5 D5⋊C8 D5⋊C8 D5⋊M5(2) kernel D5⋊M5(2) D5⋊C16 C8.F5 C20.C8 D5×C2×C8 C8×D5 C2×C40 C2×C4×D5 C4×D5 C2×Dic5 C22×D5 D5 C2×C8 C8 C2×C4 C4 C22 C1 # reps 1 2 2 2 1 4 2 2 8 4 4 8 1 2 1 2 2 8

Matrix representation of D5⋊M5(2) in GL4(𝔽241) generated by

 190 191 0 0 240 240 0 0 0 0 0 190 0 0 52 51
,
 240 50 0 0 0 1 0 0 0 0 1 0 0 0 240 240
,
 0 0 1 0 0 0 0 1 84 84 0 0 30 157 0 0
,
 1 0 0 0 0 1 0 0 0 0 240 0 0 0 0 240
`G:=sub<GL(4,GF(241))| [190,240,0,0,191,240,0,0,0,0,0,52,0,0,190,51],[240,0,0,0,50,1,0,0,0,0,1,240,0,0,0,240],[0,0,84,30,0,0,84,157,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240] >;`

D5⋊M5(2) in GAP, Magma, Sage, TeX

`D_5\rtimes M_5(2)`
`% in TeX`

`G:=Group("D5:M5(2)");`
`// GroupNames label`

`G:=SmallGroup(320,1053);`
`// by ID`

`G=gap.SmallGroup(320,1053);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,80,102,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^2=c^16=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^9>;`
`// generators/relations`

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