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## G = D10⋊10M4(2)  order 320 = 26·5

### 4th 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⋊10M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C23.2F5 — D10⋊10M4(2)
 Lower central C5 — C2×C10 — D10⋊10M4(2)
 Upper central C1 — C22 — C22×C4

Generators and relations for D1010M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=ab, dbd=a8b, dcd=c5 >

Subgroups: 762 in 190 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×18], C5, C8 [×4], C2×C4 [×2], C2×C4 [×16], C23, C23 [×8], D5 [×4], C10 [×3], C10 [×2], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×9], C24, Dic5 [×2], Dic5, C20 [×2], C20, 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 [×4], C4×D5 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10, C24.4C4, C4.F5 [×4], C2×C5⋊C8 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C23×D5, D10⋊C8 [×2], C23.2F5 [×2], C2×C4.F5 [×2], D5×C22×C4, D1010M4(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, C4.F5 [×2], C22⋊F5 [×2], C22×F5, C2×C4.F5, D5⋊M4(2), C2×C22⋊F5, D1010M4(2)

Smallest permutation representation of D1010M4(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 5)(2 4)(6 10)(7 9)(11 16)(12 15)(13 14)(17 20)(18 19)(22 30)(23 29)(24 28)(25 27)(31 37)(32 36)(33 35)(38 40)(41 45)(42 44)(46 50)(47 49)(51 56)(52 55)(53 54)(57 60)(58 59)(61 68)(62 67)(63 66)(64 65)(69 70)(71 80)(72 79)(73 78)(74 77)(75 76)
(1 14 41 59 24 71 32 65)(2 17 50 56 25 74 31 62)(3 20 49 53 26 77 40 69)(4 13 48 60 27 80 39 66)(5 16 47 57 28 73 38 63)(6 19 46 54 29 76 37 70)(7 12 45 51 30 79 36 67)(8 15 44 58 21 72 35 64)(9 18 43 55 22 75 34 61)(10 11 42 52 23 78 33 68)
(1 6)(2 5)(3 4)(7 10)(8 9)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 73)(18 72)(19 71)(20 80)(21 22)(23 30)(24 29)(25 28)(26 27)(31 38)(32 37)(33 36)(34 35)(39 40)(41 46)(42 45)(43 44)(47 50)(48 49)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 70)(60 69)```

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

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

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

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 1 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 M4(2) M4(2) F5 C2×F5 C2×F5 C22⋊F5 C4.F5 D5⋊M4(2) kernel D10⋊10M4(2) D10⋊C8 C23.2F5 C2×C4.F5 D5×C22×C4 C2×C4×D5 C22×C20 C23×D5 C4×D5 D10 C2×C10 C22×C4 C2×C4 C23 C4 C22 C2 # reps 1 2 2 2 1 4 2 2 4 4 4 1 2 1 4 4 4

Matrix representation of D1010M4(2) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 35 28 28 0 0 6 40 0 32 0 0 0 0 40 8 0 0 0 0 40 7
,
 40 0 0 0 0 0 37 1 0 0 0 0 0 0 40 6 0 13 0 0 0 1 9 0 0 0 0 0 40 0 0 0 0 0 40 1
,
 32 25 0 0 0 0 5 9 0 0 0 0 0 0 0 40 24 26 0 0 0 40 30 19 0 0 13 23 1 35 0 0 9 32 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 40 26 21 0 0 35 35 13 17 0 0 0 0 7 33 0 0 0 0 6 34

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,28,0,40,40,0,0,28,32,8,7],[40,37,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,6,1,0,0,0,0,0,9,40,40,0,0,13,0,0,1],[32,5,0,0,0,0,25,9,0,0,0,0,0,0,0,0,13,9,0,0,40,40,23,32,0,0,24,30,1,0,0,0,26,19,35,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,35,0,0,0,0,40,35,0,0,0,0,26,13,7,6,0,0,21,17,33,34] >;`

D1010M4(2) in GAP, Magma, Sage, TeX

`D_{10}\rtimes_{10}M_4(2)`
`% in TeX`

`G:=Group("D10:10M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,1094);`
`// by ID`

`G=gap.SmallGroup(320,1094);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,184,136,6278,1595]);`
`// Polycyclic`

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

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