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## G = D10⋊8M4(2)  order 320 = 26·5

### 2nd 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⋊8M4(2)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D5×C22×C4 — D10⋊8M4(2)
 Lower central C5 — C2×C10 — D10⋊8M4(2)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for D108M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c5 >

Subgroups: 718 in 190 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C23, D5, C10, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, C2×M4(2), C2×M4(2), C23×C4, C52C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C24.4C4, C2×C52C8, C4.Dic5, C2×C40, C5×M4(2), C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, D101C8, C2×C4.Dic5, C10×M4(2), D5×C22×C4, D108M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, M4(2), C22×C4, C2×D4, D10, C2×C22⋊C4, C2×M4(2), C4×D5, D20, C5⋊D4, C22×D5, C24.4C4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, D5×M4(2), C2×D10⋊C4, D108M4(2)

Smallest permutation representation of D108M4(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)(11 15)(12 14)(16 20)(17 19)(21 24)(22 23)(25 30)(26 29)(27 28)(31 34)(32 33)(35 40)(36 39)(37 38)(41 48)(42 47)(43 46)(44 45)(49 50)(51 59)(52 58)(53 57)(54 56)(61 63)(64 70)(65 69)(66 68)(71 79)(72 78)(73 77)(74 76)
(1 11 45 58 23 78 38 65)(2 12 46 59 24 79 39 66)(3 13 47 60 25 80 40 67)(4 14 48 51 26 71 31 68)(5 15 49 52 27 72 32 69)(6 16 50 53 28 73 33 70)(7 17 41 54 29 74 34 61)(8 18 42 55 30 75 35 62)(9 19 43 56 21 76 36 63)(10 20 44 57 22 77 37 64)
(11 78)(12 79)(13 80)(14 71)(15 72)(16 73)(17 74)(18 75)(19 76)(20 77)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)```

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D5 M4(2) D10 D10 C4×D5 D20 C5⋊D4 C4×D5 D5×M4(2) kernel D10⋊8M4(2) D10⋊1C8 C2×C4.Dic5 C10×M4(2) D5×C22×C4 C2×C4×D5 C22×Dic5 C23×D5 C2×C20 C2×M4(2) D10 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 4 2 2 4 2 8 4 2 4 8 8 4 8

Matrix representation of D108M4(2) in GL4(𝔽41) generated by

 34 34 0 0 7 1 0 0 0 0 40 0 0 0 0 40
,
 34 34 0 0 1 7 0 0 0 0 40 0 0 0 21 1
,
 24 1 0 0 40 17 0 0 0 0 40 37 0 0 23 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 20 40
`G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[34,1,0,0,34,7,0,0,0,0,40,21,0,0,0,1],[24,40,0,0,1,17,0,0,0,0,40,23,0,0,37,1],[1,0,0,0,0,1,0,0,0,0,1,20,0,0,0,40] >;`

D108M4(2) in GAP, Magma, Sage, TeX

`D_{10}\rtimes_8M_4(2)`
`% in TeX`

`G:=Group("D10:8M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,753);`
`// by ID`

`G=gap.SmallGroup(320,753);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,387,58,136,12550]);`
`// Polycyclic`

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

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