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## G = D4.8D10order 160 = 25·5

### 3rd non-split extension by D4 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4.8D10
 Chief series C1 — C5 — C10 — C20 — D20 — C4○D20 — D4.8D10
 Lower central C5 — C10 — C20 — D4.8D10
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for D4.8D10
G = < a,b,c,d | a4=b2=1, c10=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c9 >

Subgroups: 184 in 62 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4○D8, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, D4.8D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4○D8, C5⋊D4, C22×D5, C2×C5⋊D4, D4.8D10

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

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

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

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

34 conjugacy classes

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

34 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 C2 C2 C2 D4 D4 D5 D10 D10 D10 C4○D8 C5⋊D4 C5⋊D4 D4.8D10 kernel D4.8D10 C2×C5⋊2C8 D4⋊D5 D4.D5 Q8⋊D5 C5⋊Q16 C4○D20 C5×C4○D4 C20 C2×C10 C4○D4 C2×C4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4

Matrix representation of D4.8D10 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 18 0 0 9 40
,
 24 1 0 0 40 17 0 0 0 0 0 30 0 0 26 0
,
 7 7 0 0 34 40 0 0 0 0 32 0 0 0 0 32
,
 7 7 0 0 40 34 0 0 0 0 32 0 0 0 1 9
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,9,0,0,18,40],[24,40,0,0,1,17,0,0,0,0,0,26,0,0,30,0],[7,34,0,0,7,40,0,0,0,0,32,0,0,0,0,32],[7,40,0,0,7,34,0,0,0,0,32,1,0,0,0,9] >;`

D4.8D10 in GAP, Magma, Sage, TeX

`D_4._8D_{10}`
`% in TeX`

`G:=Group("D4.8D10");`
`// GroupNames label`

`G:=SmallGroup(160,171);`
`// by ID`

`G=gap.SmallGroup(160,171);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,579,159,69,4613]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^10=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^9>;`
`// generators/relations`

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