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## G = D10.Q8order 160 = 25·5

### 2nd non-split extension by D10 of Q8 acting via Q8/C4=C2

Aliases: C8.1F5, C40.1C4, D10.2Q8, Dic5.11D4, C52C8.4C4, (C8×D5).4C2, C2.7(C4⋊F5), C10.4(C4⋊C4), C4.11(C2×F5), C52(C8.C4), C4.F5.2C2, C20.11(C2×C4), (C4×D5).28C22, SmallGroup(160,71)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D10.Q8
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — D10.Q8
 Lower central C5 — C10 — C20 — D10.Q8
 Upper central C1 — C2 — C4 — C8

Generators and relations for D10.Q8
G = < a,b,c,d | a10=b2=1, c4=a5, d2=a-1bc2, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a7b, dcd-1=a5c3 >

Character table of D10.Q8

 class 1 2A 2B 4A 4B 4C 5 8A 8B 8C 8D 8E 8F 8G 8H 10 20A 20B 40A 40B 40C 40D size 1 1 10 2 5 5 4 2 2 10 10 20 20 20 20 4 4 4 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 trivial ρ2 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 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 linear of order 2 ρ5 1 1 -1 1 -1 -1 1 -1 -1 1 1 i -i -i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 1 -1 -1 1 -1 -1 1 1 -i i i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 -1 1 -1 -1 1 1 1 -1 -1 -i -i i i 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 1 -1 -1 1 1 1 -1 -1 i i -i -i 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 2 -2 0 0 -2i 2i 2 -√2 √2 -√-2 √-2 0 0 0 0 -2 0 0 √2 -√2 -√2 √2 complex lifted from C8.C4 ρ12 2 -2 0 0 2i -2i 2 -√2 √2 √-2 -√-2 0 0 0 0 -2 0 0 √2 -√2 -√2 √2 complex lifted from C8.C4 ρ13 2 -2 0 0 2i -2i 2 √2 -√2 -√-2 √-2 0 0 0 0 -2 0 0 -√2 √2 √2 -√2 complex lifted from C8.C4 ρ14 2 -2 0 0 -2i 2i 2 √2 -√2 √-2 -√-2 0 0 0 0 -2 0 0 -√2 √2 √2 -√2 complex lifted from C8.C4 ρ15 4 4 0 4 0 0 -1 -4 -4 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ16 4 4 0 4 0 0 -1 4 4 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ17 4 4 0 -4 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 -√-5 -√-5 √-5 √-5 complex lifted from C4⋊F5 ρ18 4 4 0 -4 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 √-5 √-5 -√-5 -√-5 complex lifted from C4⋊F5 ρ19 4 -4 0 0 0 0 -1 -2√2 2√2 0 0 0 0 0 0 1 -√-5 √-5 ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 complex faithful ρ20 4 -4 0 0 0 0 -1 2√2 -2√2 0 0 0 0 0 0 1 √-5 -√-5 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 complex faithful ρ21 4 -4 0 0 0 0 -1 -2√2 2√2 0 0 0 0 0 0 1 √-5 -√-5 ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 complex faithful ρ22 4 -4 0 0 0 0 -1 2√2 -2√2 0 0 0 0 0 0 1 -√-5 √-5 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 complex faithful

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

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

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

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

D10.Q8 is a maximal subgroup of
C16.F5  C80.2C4  D8.F5  Q16.F5  (C8×D5).C4  M4(2).1F5  D85F5  SD162F5  Q165F5  D10.2Dic6  C24.1F5
D10.Q8 is a maximal quotient of
C401C8  C20.26M4(2)  C20.10C42  D10.2Dic6  C24.1F5

Matrix representation of D10.Q8 in GL4(𝔽7) generated by

 6 0 4 2 5 1 0 0 6 3 3 4 3 5 1 5
,
 6 0 4 2 0 6 1 4 0 3 3 1 0 1 3 6
,
 6 3 0 2 1 6 1 1 2 2 2 3 3 4 2 6
,
 1 1 6 1 0 2 6 4 3 5 4 6 5 4 6 0
`G:=sub<GL(4,GF(7))| [6,5,6,3,0,1,3,5,4,0,3,1,2,0,4,5],[6,0,0,0,0,6,3,1,4,1,3,3,2,4,1,6],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[1,0,3,5,1,2,5,4,6,6,4,6,1,4,6,0] >;`

D10.Q8 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(160,71);`
`// by ID`

`G=gap.SmallGroup(160,71);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,151,86,579,69,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=1,c^4=a^5,d^2=a^-1*b*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=a^5*c^3>;`
`// generators/relations`

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