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## G = Q8.D10order 160 = 25·5

### 5th non-split extension by Q8 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Q8.D10
 Chief series C1 — C5 — C10 — C20 — C4×D5 — Q8⋊2D5 — Q8.D10
 Lower central C5 — C10 — C20 — Q8.D10
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q8.D10
G = < a,b,c,d | a4=d2=1, b2=c10=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=a2c9 >

Subgroups: 232 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, D5, C10, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, D10, C4○D8, C52C8, C40, C4×D5, C4×D5, D20, D20, C5×Q8, C8×D5, D40, Q8⋊D5, C5×Q16, Q82D5, Q8.D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4○D8, C22×D5, D4×D5, Q8.D10

Character table of Q8.D10

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 10A 10B 20A 20B 20C 20D 20E 20F 40A 40B 40C 40D size 1 1 10 20 20 2 4 4 5 5 2 2 2 2 10 10 2 2 4 4 8 8 8 8 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ7 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ9 2 2 2 0 0 -2 0 0 -2 -2 2 2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 0 -2 0 0 2 2 2 2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 0 0 0 2 2 2 0 0 -1-√5/2 -1+√5/2 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ12 2 2 0 0 0 2 -2 -2 0 0 -1-√5/2 -1+√5/2 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D10 ρ13 2 2 0 0 0 2 2 -2 0 0 -1+√5/2 -1-√5/2 -2 -2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ14 2 2 0 0 0 2 -2 -2 0 0 -1+√5/2 -1-√5/2 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D10 ρ15 2 2 0 0 0 2 -2 2 0 0 -1-√5/2 -1+√5/2 -2 -2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ16 2 2 0 0 0 2 -2 2 0 0 -1+√5/2 -1-√5/2 -2 -2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ17 2 2 0 0 0 2 2 2 0 0 -1+√5/2 -1-√5/2 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ18 2 2 0 0 0 2 2 -2 0 0 -1-√5/2 -1+√5/2 -2 -2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ19 2 -2 0 0 0 0 0 0 2i -2i 2 2 √2 -√2 √-2 -√-2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 complex lifted from C4○D8 ρ20 2 -2 0 0 0 0 0 0 -2i 2i 2 2 √2 -√2 -√-2 √-2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 complex lifted from C4○D8 ρ21 2 -2 0 0 0 0 0 0 -2i 2i 2 2 -√2 √2 √-2 -√-2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 complex lifted from C4○D8 ρ22 2 -2 0 0 0 0 0 0 2i -2i 2 2 -√2 √2 -√-2 √-2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 complex lifted from C4○D8 ρ23 4 4 0 0 0 -4 0 0 0 0 -1+√5 -1-√5 0 0 0 0 -1-√5 -1+√5 1-√5 1+√5 0 0 0 0 0 0 0 0 orthogonal lifted from D4×D5 ρ24 4 4 0 0 0 -4 0 0 0 0 -1-√5 -1+√5 0 0 0 0 -1+√5 -1-√5 1+√5 1-√5 0 0 0 0 0 0 0 0 orthogonal lifted from D4×D5 ρ25 4 -4 0 0 0 0 0 0 0 0 -1-√5 -1+√5 -2√2 2√2 0 0 1-√5 1+√5 0 0 0 0 0 0 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 orthogonal faithful ρ26 4 -4 0 0 0 0 0 0 0 0 -1-√5 -1+√5 2√2 -2√2 0 0 1-√5 1+√5 0 0 0 0 0 0 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 orthogonal faithful ρ27 4 -4 0 0 0 0 0 0 0 0 -1+√5 -1-√5 -2√2 2√2 0 0 1+√5 1-√5 0 0 0 0 0 0 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 orthogonal faithful ρ28 4 -4 0 0 0 0 0 0 0 0 -1+√5 -1-√5 2√2 -2√2 0 0 1+√5 1-√5 0 0 0 0 0 0 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 orthogonal faithful

Smallest permutation representation of Q8.D10
On 80 points
Generators in S80
```(1 26 11 36)(2 37 12 27)(3 28 13 38)(4 39 14 29)(5 30 15 40)(6 21 16 31)(7 32 17 22)(8 23 18 33)(9 34 19 24)(10 25 20 35)(41 73 51 63)(42 64 52 74)(43 75 53 65)(44 66 54 76)(45 77 55 67)(46 68 56 78)(47 79 57 69)(48 70 58 80)(49 61 59 71)(50 72 60 62)
(1 44 11 54)(2 67 12 77)(3 46 13 56)(4 69 14 79)(5 48 15 58)(6 71 16 61)(7 50 17 60)(8 73 18 63)(9 52 19 42)(10 75 20 65)(21 59 31 49)(22 72 32 62)(23 41 33 51)(24 74 34 64)(25 43 35 53)(26 76 36 66)(27 45 37 55)(28 78 38 68)(29 47 39 57)(30 80 40 70)
(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 43)(2 42)(3 41)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)```

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

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

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

Matrix representation of Q8.D10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 37 0 0 21 40
,
 1 0 0 0 0 1 0 0 0 0 32 36 0 0 0 9
,
 1 6 0 0 35 6 0 0 0 0 11 19 0 0 13 30
,
 1 0 0 0 35 40 0 0 0 0 0 34 0 0 35 0
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,21,0,0,37,40],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,36,9],[1,35,0,0,6,6,0,0,0,0,11,13,0,0,19,30],[1,35,0,0,0,40,0,0,0,0,0,35,0,0,34,0] >;`

Q8.D10 in GAP, Magma, Sage, TeX

`Q_8.D_{10}`
`% in TeX`

`G:=Group("Q8.D10");`
`// GroupNames label`

`G:=SmallGroup(160,140);`
`// by ID`

`G=gap.SmallGroup(160,140);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,362,116,86,297,159,69,4613]);`
`// Polycyclic`

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

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