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## G = D4.9D10order 160 = 25·5

### 4th non-split extension by D4 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4.9D10
 Chief series C1 — C5 — C10 — C20 — Dic10 — C2×Dic10 — D4.9D10
 Lower central C5 — C10 — C20 — D4.9D10
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for D4.9D10
G = < a,b,c,d | a4=b2=c10=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 160 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C8.C22, C52C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4.Dic5, D4.D5, C5⋊Q16, C2×Dic10, C5×C4○D4, D4.9D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8.C22, C5⋊D4, C22×D5, C2×C5⋊D4, D4.9D10

Smallest permutation representation of D4.9D10
On 80 points
Generators in S80
```(1 41 33 15)(2 42 34 16)(3 43 35 17)(4 44 36 18)(5 45 37 19)(6 46 38 20)(7 47 39 11)(8 48 40 12)(9 49 31 13)(10 50 32 14)(21 58 65 78)(22 59 66 79)(23 60 67 80)(24 51 68 71)(25 52 69 72)(26 53 70 73)(27 54 61 74)(28 55 62 75)(29 56 63 76)(30 57 64 77)
(1 15)(2 42)(3 17)(4 44)(5 19)(6 46)(7 11)(8 48)(9 13)(10 50)(12 40)(14 32)(16 34)(18 36)(20 38)(21 65)(23 67)(25 69)(27 61)(29 63)(31 49)(33 41)(35 43)(37 45)(39 47)(51 71)(53 73)(55 75)(57 77)(59 79)
(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 76 33 56)(2 75 34 55)(3 74 35 54)(4 73 36 53)(5 72 37 52)(6 71 38 51)(7 80 39 60)(8 79 40 59)(9 78 31 58)(10 77 32 57)(11 23 47 67)(12 22 48 66)(13 21 49 65)(14 30 50 64)(15 29 41 63)(16 28 42 62)(17 27 43 61)(18 26 44 70)(19 25 45 69)(20 24 46 68)```

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

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

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

31 conjugacy classes

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

31 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 C5⋊D4 C5⋊D4 C8.C22 D4.9D10 kernel D4.9D10 C4.Dic5 D4.D5 C5⋊Q16 C2×Dic10 C5×C4○D4 C20 C2×C10 C4○D4 C2×C4 D4 Q8 C4 C22 C5 C1 # reps 1 1 2 2 1 1 1 1 2 2 2 2 4 4 1 4

Matrix representation of D4.9D10 in GL4(𝔽41) generated by

 1 0 0 28 0 1 13 13 3 3 40 0 38 0 0 40
,
 1 0 0 28 0 1 13 13 0 0 40 0 0 0 0 40
,
 25 14 18 21 29 13 5 38 39 40 14 27 7 40 14 30
,
 25 36 29 10 27 16 22 31 37 37 30 0 24 28 36 11
`G:=sub<GL(4,GF(41))| [1,0,3,38,0,1,3,0,0,13,40,0,28,13,0,40],[1,0,0,0,0,1,0,0,0,13,40,0,28,13,0,40],[25,29,39,7,14,13,40,40,18,5,14,14,21,38,27,30],[25,27,37,24,36,16,37,28,29,22,30,36,10,31,0,11] >;`

D4.9D10 in GAP, Magma, Sage, TeX

`D_4._9D_{10}`
`% in TeX`

`G:=Group("D4.9D10");`
`// GroupNames label`

`G:=SmallGroup(160,172);`
`// by ID`

`G=gap.SmallGroup(160,172);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,188,579,159,69,4613]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^10=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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