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## G = D10⋊9M4(2)  order 320 = 26·5

### 3rd semidirect product of D10 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10⋊9M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — D10⋊C8 — D10⋊9M4(2)
 Lower central C5 — C2×C10 — D10⋊9M4(2)
 Upper central C1 — C22 — C22×C4

Generators and relations for D109M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, bd=db, dcd=c5 >

Subgroups: 762 in 190 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C22, C22 [×2], C22 [×18], C5, C8 [×4], C2×C4 [×2], C2×C4 [×16], C23, C23 [×8], D5 [×4], C10, C10 [×2], C10 [×2], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×9], C24, Dic5 [×4], C20 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8 [×4], C2×M4(2) [×2], C23×C4, C5⋊C8 [×4], C4×D5 [×8], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10, C24.4C4, C2×C5⋊C8 [×4], C22.F5 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C23×D5, D10⋊C8 [×4], C2×C22.F5 [×2], D5×C22×C4, D109M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×M4(2) [×2], C2×F5 [×3], C24.4C4, C22⋊F5 [×2], C22×F5, D5⋊M4(2) [×2], C2×C22⋊F5, D109M4(2)

Smallest permutation representation of D109M4(2)
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 10)(2 9)(3 8)(4 7)(5 6)(12 20)(13 19)(14 18)(15 17)(21 22)(23 30)(24 29)(25 28)(26 27)(31 32)(33 40)(34 39)(35 38)(36 37)(41 50)(42 49)(43 48)(44 47)(45 46)(52 60)(53 59)(54 58)(55 57)(61 63)(64 70)(65 69)(66 68)(71 75)(72 74)(76 80)(77 79)
(1 76 41 70 22 19 32 54)(2 73 50 63 23 16 31 57)(3 80 49 66 24 13 40 60)(4 77 48 69 25 20 39 53)(5 74 47 62 26 17 38 56)(6 71 46 65 27 14 37 59)(7 78 45 68 28 11 36 52)(8 75 44 61 29 18 35 55)(9 72 43 64 30 15 34 58)(10 79 42 67 21 12 33 51)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 21)(31 50)(32 41)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 C4 D4 M4(2) F5 C2×F5 C2×F5 C22⋊F5 D5⋊M4(2) kernel D10⋊9M4(2) D10⋊C8 C2×C22.F5 D5×C22×C4 C2×C4×D5 C22×C20 C23×D5 C2×Dic5 D10 C22×C4 C2×C4 C23 C22 C2 # reps 1 4 2 1 4 2 2 4 8 1 2 1 4 8

Matrix representation of D109M4(2) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 40 0 0 1 1 1 1 0 0 40 0 0 0 0 0 0 40 0 0
,
 40 0 0 0 0 0 33 1 0 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40 0 0 0
,
 33 2 0 0 0 0 34 8 0 0 0 0 0 0 6 12 11 1 0 0 40 30 29 35 0 0 40 5 11 10 0 0 6 5 36 35
,
 1 0 0 0 0 0 8 40 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

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,40,0,0,0,1,0,0,0,0,40,1,0,0],[40,33,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0],[33,34,0,0,0,0,2,8,0,0,0,0,0,0,6,40,40,6,0,0,12,30,5,5,0,0,11,29,11,36,0,0,1,35,10,35],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

D109M4(2) in GAP, Magma, Sage, TeX

`D_{10}\rtimes_9M_4(2)`
`% in TeX`

`G:=Group("D10:9M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,1093);`
`// by ID`

`G=gap.SmallGroup(320,1093);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,232,758,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=c^5>;`
`// generators/relations`

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