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## G = Q8○D8order 64 = 26

### Central product of Q8 and D8

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — Q8○D8
 Chief series C1 — C2 — C4 — C2×C4 — C4○D4 — 2- 1+4 — Q8○D8
 Lower central C1 — C2 — C4 — Q8○D8
 Upper central C1 — C2 — C4○D4 — Q8○D8
 Jennings C1 — C2 — C2 — C4 — Q8○D8

Generators and relations for Q8○D8
G = < a,b,c,d | a4=d2=1, b2=c4=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c3 >

Subgroups: 173 in 124 conjugacy classes, 79 normal (9 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, Q8○D8
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, Q8○D8

Character table of Q8○D8

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E size 1 1 2 2 2 4 4 2 2 2 2 4 4 4 4 4 4 2 2 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 -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 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 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 linear of order 2 ρ9 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 2 -2 2 2 0 0 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 0 0 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 -2 0 0 2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 -2 2 0 0 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 symplectic faithful, Schur index 2 ρ22 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Q8○D8
On 32 points
Generators in S32
```(1 26 5 30)(2 27 6 31)(3 28 7 32)(4 29 8 25)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)
(1 13 5 9)(2 14 6 10)(3 15 7 11)(4 16 8 12)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)
(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)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)(25 26)(27 32)(28 31)(29 30)```

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

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

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

Matrix representation of Q8○D8 in GL4(𝔽7) generated by

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

Q8○D8 in GAP, Magma, Sage, TeX

`Q_8\circ D_8`
`% in TeX`

`G:=Group("Q8oD8");`
`// GroupNames label`

`G:=SmallGroup(64,259);`
`// by ID`

`G=gap.SmallGroup(64,259);`
`# by ID`

`G:=PCGroup([6,-2,2,2,2,-2,-2,192,217,199,255,1444,730,88]);`
`// Polycyclic`

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

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