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## G = Q8⋊2D5order 80 = 24·5

### The semidirect product of Q8 and D5 acting through Inn(Q8)

Aliases: Q82D5, Q8Dic5, D204C2, C4.7D10, C20.7C22, C10.8C23, D10.3C22, Dic5.9C22, (C4×D5)⋊3C2, C53(C4○D4), (C5×Q8)⋊3C2, C2.9(C22×D5), SmallGroup(80,42)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q82D5
G = < a,b,c,d | a4=c5=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Character table of Q82D5

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 10A 10B 20A 20B 20C 20D 20E 20F size 1 1 10 10 10 2 2 2 5 5 2 2 2 2 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 trivial ρ2 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 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 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 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 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 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 linear of order 2 ρ9 2 2 0 0 0 -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 orthogonal lifted from D10 ρ10 2 2 0 0 0 -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 orthogonal lifted from D10 ρ11 2 2 0 0 0 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 orthogonal lifted from D10 ρ12 2 2 0 0 0 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 orthogonal lifted from D10 ρ13 2 2 0 0 0 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 orthogonal lifted from D5 ρ14 2 2 0 0 0 -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 orthogonal lifted from D10 ρ15 2 2 0 0 0 -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 orthogonal lifted from D10 ρ16 2 2 0 0 0 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 orthogonal lifted from D5 ρ17 2 -2 0 0 0 0 0 0 -2i 2i 2 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 0 0 0 0 0 0 2i -2i 2 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 4 -4 0 0 0 0 0 0 0 0 -1+√5 -1-√5 1+√5 1-√5 0 0 0 0 0 0 orthogonal faithful ρ20 4 -4 0 0 0 0 0 0 0 0 -1-√5 -1+√5 1-√5 1+√5 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of Q82D5
On 40 points
Generators in S40
```(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)
(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)
(1 5)(2 4)(7 10)(8 9)(11 16)(12 20)(13 19)(14 18)(15 17)(22 25)(23 24)(27 30)(28 29)(31 36)(32 40)(33 39)(34 38)(35 37)```

`G:=sub<Sym(40)| (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35), (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), (1,5)(2,4)(7,10)(8,9)(11,16)(12,20)(13,19)(14,18)(15,17)(22,25)(23,24)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)>;`

`G:=Group( (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35), (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), (1,5)(2,4)(7,10)(8,9)(11,16)(12,20)(13,19)(14,18)(15,17)(22,25)(23,24)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37) );`

`G=PermutationGroup([(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,29,9,24),(2,30,10,25),(3,26,6,21),(4,27,7,22),(5,28,8,23),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35)], [(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)], [(1,5),(2,4),(7,10),(8,9),(11,16),(12,20),(13,19),(14,18),(15,17),(22,25),(23,24),(27,30),(28,29),(31,36),(32,40),(33,39),(34,38),(35,37)])`

Matrix representation of Q82D5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 0 1 0 0 40 0
,
 40 0 0 0 0 40 0 0 0 0 9 0 0 0 0 32
,
 0 1 0 0 40 6 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 40
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,40] >;`

Q82D5 in GAP, Magma, Sage, TeX

`Q_8\rtimes_2D_5`
`% in TeX`

`G:=Group("Q8:2D5");`
`// GroupNames label`

`G:=SmallGroup(80,42);`
`// by ID`

`G=gap.SmallGroup(80,42);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,46,182,97,42,1604]);`
`// Polycyclic`

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

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