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## G = Q8.D8order 128 = 27

### 1st non-split extension by Q8 of D8 acting via D8/D4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — Q8.D8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — Q8⋊6D4 — Q8.D8
 Lower central C1 — C22 — C42 — Q8.D8
 Upper central C1 — C22 — C42 — Q8.D8
 Jennings C1 — C22 — C22 — C42 — Q8.D8

Generators and relations for Q8.D8
G = < a,b,c,d | a4=c8=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a-1b, dcd-1=a2c-1 >

Subgroups: 328 in 125 conjugacy classes, 36 normal (32 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×6], C22, C22 [×9], C8 [×3], C2×C4 [×3], C2×C4 [×10], D4 [×14], Q8 [×2], Q8 [×3], C23 [×3], C42, C42, C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×3], Q16 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C2×Q8, C4○D4 [×4], C4×C8, D4⋊C4 [×3], Q8⋊C4, C4⋊C8 [×2], C2.D8, C4×D4, C4×D4, C4×Q8, C4⋊D4 [×3], C41D4, C41D4, C4⋊Q8, C2×Q16, C2×C4○D4, D4⋊C8, Q8⋊C8, C4.D8, C42Q16, D4⋊Q8, C4.4D8, Q86D4, Q8.D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C4○D8, C8⋊C22, C8.C22, C22⋊D8, D4.7D4, D4.9D4, 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 4K 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 8 8 8 2 2 2 2 4 4 4 4 4 8 16 4 4 4 4 8 8 8 8 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 2 0 0 2 -2 2 -2 -2 0 -2 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 0 -2 2 -2 2 -2 2 0 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 0 0 2 -2 2 -2 -2 0 2 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 -2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 0 -2 2 -2 2 -2 -2 0 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 0 2 -2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 2 -2 0 0 0 0 2 0 -2 0 -2 0 2 0 0 0 √2 -√2 √2 -√2 0 √2 0 -√2 orthogonal lifted from D8 ρ16 2 -2 2 -2 0 0 0 0 2 0 -2 0 -2 0 2 0 0 0 -√2 √2 -√2 √2 0 -√2 0 √2 orthogonal lifted from D8 ρ17 2 -2 2 -2 0 0 0 0 2 0 -2 0 2 0 -2 0 0 0 -√2 √2 -√2 √2 0 √2 0 -√2 orthogonal lifted from D8 ρ18 2 -2 2 -2 0 0 0 0 2 0 -2 0 2 0 -2 0 0 0 √2 -√2 √2 -√2 0 -√2 0 √2 orthogonal lifted from D8 ρ19 2 2 -2 -2 0 0 0 2 0 -2 0 0 0 2i 0 -2i 0 0 √-2 √-2 -√-2 -√-2 √2 0 -√2 0 complex lifted from C4○D8 ρ20 2 2 -2 -2 0 0 0 2 0 -2 0 0 0 -2i 0 2i 0 0 -√-2 -√-2 √-2 √-2 √2 0 -√2 0 complex lifted from C4○D8 ρ21 2 2 -2 -2 0 0 0 2 0 -2 0 0 0 -2i 0 2i 0 0 √-2 √-2 -√-2 -√-2 -√2 0 √2 0 complex lifted from C4○D8 ρ22 2 2 -2 -2 0 0 0 2 0 -2 0 0 0 2i 0 -2i 0 0 -√-2 -√-2 √-2 √-2 -√2 0 √2 0 complex lifted from C4○D8 ρ23 4 -4 4 -4 0 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ24 4 4 -4 -4 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 complex lifted from D4.9D4 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 complex lifted from D4.9D4

Smallest permutation representation of Q8.D8
On 64 points
Generators in S64
```(1 34 55 61)(2 35 56 62)(3 36 49 63)(4 37 50 64)(5 38 51 57)(6 39 52 58)(7 40 53 59)(8 33 54 60)(9 29 17 41)(10 30 18 42)(11 31 19 43)(12 32 20 44)(13 25 21 45)(14 26 22 46)(15 27 23 47)(16 28 24 48)
(1 15 55 23)(2 28 56 48)(3 17 49 9)(4 42 50 30)(5 11 51 19)(6 32 52 44)(7 21 53 13)(8 46 54 26)(10 37 18 64)(12 58 20 39)(14 33 22 60)(16 62 24 35)(25 59 45 40)(27 34 47 61)(29 63 41 36)(31 38 43 57)
(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)
(1 54 55 8)(2 7 56 53)(3 52 49 6)(4 5 50 51)(9 20 17 12)(10 11 18 19)(13 24 21 16)(14 15 22 23)(25 28 45 48)(26 47 46 27)(29 32 41 44)(30 43 42 31)(33 61 60 34)(35 59 62 40)(36 39 63 58)(37 57 64 38)```

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

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

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

Matrix representation of Q8.D8 in GL4(𝔽17) generated by

 0 1 0 0 16 0 0 0 0 0 1 0 0 0 0 1
,
 13 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 12 5 0 0 12 12 0 0 0 0 6 6 0 0 14 0
,
 12 5 0 0 5 5 0 0 0 0 6 6 0 0 14 11
`G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,5,12,0,0,0,0,6,14,0,0,6,0],[12,5,0,0,5,5,0,0,0,0,6,14,0,0,6,11] >;`

Q8.D8 in GAP, Magma, Sage, TeX

`Q_8.D_8`
`% in TeX`

`G:=Group("Q8.D8");`
`// GroupNames label`

`G:=SmallGroup(128,373);`
`// by ID`

`G=gap.SmallGroup(128,373);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,352,1123,570,521,136,2804,1411,718,172]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^8=1,b^2=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

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