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G = Q8⋊D5order 80 = 24·5

The semidirect product of Q8 and D5 acting via D5/C5=C2

Aliases: Q8⋊D5, C53SD16, C4.3D10, C10.9D4, D20.2C2, C20.3C22, C52C83C2, (C5×Q8)⋊1C2, C2.6(C5⋊D4), SmallGroup(80,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Q8⋊D5
 Chief series C1 — C5 — C10 — C20 — D20 — Q8⋊D5
 Lower central C5 — C10 — C20 — Q8⋊D5
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q8⋊D5
G = < a,b,c,d | a4=c5=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

Character table of Q8⋊D5

 class 1 2A 2B 4A 4B 5A 5B 8A 8B 10A 10B 20A 20B 20C 20D 20E 20F size 1 1 20 2 4 2 2 10 10 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 trivial ρ2 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 linear of order 2 ρ4 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 0 -2 0 2 2 0 0 2 2 -2 0 0 0 -2 0 orthogonal lifted from D4 ρ6 2 2 0 2 -2 -1+√5/2 -1-√5/2 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 orthogonal lifted from D10 ρ7 2 2 0 2 -2 -1-√5/2 -1+√5/2 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 orthogonal lifted from D10 ρ8 2 2 0 2 2 -1+√5/2 -1-√5/2 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 orthogonal lifted from D5 ρ9 2 2 0 2 2 -1-√5/2 -1+√5/2 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 orthogonal lifted from D5 ρ10 2 2 0 -2 0 -1+√5/2 -1-√5/2 0 0 -1-√5/2 -1+√5/2 1+√5/2 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 1-√5/2 -ζ53+ζ52 complex lifted from C5⋊D4 ρ11 2 2 0 -2 0 -1-√5/2 -1+√5/2 0 0 -1+√5/2 -1-√5/2 1-√5/2 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 1+√5/2 ζ54-ζ5 complex lifted from C5⋊D4 ρ12 2 2 0 -2 0 -1-√5/2 -1+√5/2 0 0 -1+√5/2 -1-√5/2 1-√5/2 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 1+√5/2 -ζ54+ζ5 complex lifted from C5⋊D4 ρ13 2 2 0 -2 0 -1+√5/2 -1-√5/2 0 0 -1-√5/2 -1+√5/2 1+√5/2 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 1-√5/2 ζ53-ζ52 complex lifted from C5⋊D4 ρ14 2 -2 0 0 0 2 2 -√-2 √-2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ15 2 -2 0 0 0 2 2 √-2 -√-2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ16 4 -4 0 0 0 -1-√5 -1+√5 0 0 1-√5 1+√5 0 0 0 0 0 0 orthogonal faithful, Schur index 2 ρ17 4 -4 0 0 0 -1+√5 -1-√5 0 0 1+√5 1-√5 0 0 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of Q8⋊D5
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)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 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)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37)>;`

`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)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37) );`

`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),(21,31),(22,35),(23,34),(24,33),(25,32),(26,36),(27,40),(28,39),(29,38),(30,37)])`

Q8⋊D5 is a maximal subgroup of
D5×SD16  D40⋊C2  Q16⋊D5  Q8.D10  C20.C23  D4⋊D10  D4.8D10  C30.D4  Dic6⋊D5  Q82D15  Q8⋊D15  Q8⋊D25  D20.D5  C524SD16  C5210SD16
Q8⋊D5 is a maximal quotient of
C20.Q8  D206C4  Q8⋊Dic5  C30.D4  Dic6⋊D5  Q82D15  Q8⋊D25  D20.D5  C524SD16  C5210SD16

Matrix representation of Q8⋊D5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 0 1 0 0 40 0
,
 18 35 0 0 6 23 0 0 0 0 15 15 0 0 15 26
,
 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],[18,6,0,0,35,23,0,0,0,0,15,15,0,0,15,26],[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] >;`

Q8⋊D5 in GAP, Magma, Sage, TeX

`Q_8\rtimes D_5`
`% in TeX`

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

`G:=SmallGroup(80,17);`
`// by ID`

`G=gap.SmallGroup(80,17);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,61,46,182,97,42,1604]);`
`// 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,d*b*d=a^-1*b,d*c*d=c^-1>;`
`// generators/relations`

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