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## G = D8.Q8order 128 = 27

### 1st non-split extension by D8 of Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — D8.Q8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×D8 — C4×D8 — D8.Q8
 Lower central C1 — C2 — C4 — C2×C8 — D8.Q8
 Upper central C1 — C22 — C42 — C4×C8 — D8.Q8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8.Q8

Generators and relations for D8.Q8
G = < a,b,c,d | a8=b2=c4=1, d2=a4c2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=a4c-1 >

Subgroups: 180 in 68 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C8 [×2], C8, C2×C4 [×3], C2×C4 [×5], D4 [×3], C23, C16 [×2], C42, C22⋊C4, C4⋊C4 [×5], C2×C8 [×2], D8 [×2], D8, C22×C4, C2×D4, C4×C8, D4⋊C4, C4.Q8, C2.D8 [×3], C2×C16 [×2], C4×D4, C42.C2, C2×D8, C2.D16 [×2], C4⋊C16, C163C4, C164C4, C4×D8, C8.5Q8, D8.Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D8 [×2], C2×D4, C2×Q8, C4○D4, C22⋊Q8, C2×D8, C8.C22, D4⋊Q8, C4○D16, C16⋊C22, D8.Q8

Character table of D8.Q8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 8 8 2 2 2 2 4 8 8 16 16 2 2 2 2 4 4 4 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 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 -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 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 -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 -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 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 -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 1 1 1 linear of order 2 ρ9 2 2 2 2 0 0 2 -2 -2 2 -2 0 0 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ11 2 2 2 2 0 0 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 -√2 √2 √2 √2 -√2 orthogonal lifted from D8 ρ13 2 2 2 2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 √2 -√2 -√2 -√2 √2 orthogonal lifted from D8 ρ14 2 2 2 2 0 0 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ15 2 -2 -2 2 2 -2 2 0 0 -2 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 -2 2 -2 2 2 0 0 -2 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 -2 2 0 0 2 0 0 -2 0 -2i 2i 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 0 0 2 0 0 -2 0 2i -2i 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 -2 2 -2 0 0 0 -2i 2i 0 0 0 0 0 0 -√2 √2 -√2 √2 √-2 -√-2 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 -ζ165+ζ163 ζ1615+ζ169 ζ1615-ζ169 -ζ1615+ζ169 ζ165-ζ163 complex lifted from C4○D16 ρ20 2 -2 2 -2 0 0 0 2i -2i 0 0 0 0 0 0 -√2 √2 -√2 √2 -√-2 √-2 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 -ζ165+ζ163 ζ167+ζ16 ζ1615-ζ169 -ζ1615+ζ169 ζ165-ζ163 complex lifted from C4○D16 ρ21 2 -2 2 -2 0 0 0 -2i 2i 0 0 0 0 0 0 √2 -√2 √2 -√2 -√-2 √-2 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 -ζ1615+ζ169 ζ165+ζ163 -ζ165+ζ163 ζ165-ζ163 ζ1615-ζ169 complex lifted from C4○D16 ρ22 2 -2 2 -2 0 0 0 2i -2i 0 0 0 0 0 0 √2 -√2 √2 -√2 √-2 -√-2 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 -ζ1615+ζ169 ζ1613+ζ1611 -ζ165+ζ163 ζ165-ζ163 ζ1615-ζ169 complex lifted from C4○D16 ρ23 2 -2 2 -2 0 0 0 -2i 2i 0 0 0 0 0 0 √2 -√2 √2 -√2 -√-2 √-2 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 ζ1615-ζ169 ζ1613+ζ1611 ζ165-ζ163 -ζ165+ζ163 -ζ1615+ζ169 complex lifted from C4○D16 ρ24 2 -2 2 -2 0 0 0 2i -2i 0 0 0 0 0 0 √2 -√2 √2 -√2 √-2 -√-2 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 ζ1615-ζ169 ζ165+ζ163 ζ165-ζ163 -ζ165+ζ163 -ζ1615+ζ169 complex lifted from C4○D16 ρ25 2 -2 2 -2 0 0 0 -2i 2i 0 0 0 0 0 0 -√2 √2 -√2 √2 √-2 -√-2 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 ζ165-ζ163 ζ167+ζ16 -ζ1615+ζ169 ζ1615-ζ169 -ζ165+ζ163 complex lifted from C4○D16 ρ26 2 -2 2 -2 0 0 0 2i -2i 0 0 0 0 0 0 -√2 √2 -√2 √2 -√-2 √-2 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 ζ165-ζ163 ζ1615+ζ169 -ζ1615+ζ169 ζ1615-ζ169 -ζ165+ζ163 complex lifted from C4○D16 ρ27 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C16⋊C22 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C16⋊C22 ρ29 4 -4 -4 4 0 0 -4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

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

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

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

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

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

 11 11 0 0 3 0 0 0 0 0 1 0 0 0 0 1
,
 11 11 0 0 3 6 0 0 0 0 16 0 0 0 0 16
,
 4 0 0 0 0 4 0 0 0 0 15 13 0 0 14 2
,
 15 5 0 0 13 2 0 0 0 0 4 6 0 0 0 13
`G:=sub<GL(4,GF(17))| [11,3,0,0,11,0,0,0,0,0,1,0,0,0,0,1],[11,3,0,0,11,6,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,15,14,0,0,13,2],[15,13,0,0,5,2,0,0,0,0,4,0,0,0,6,13] >;`

D8.Q8 in GAP, Magma, Sage, TeX

`D_8.Q_8`
`% in TeX`

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

`G:=SmallGroup(128,960);`
`// by ID`

`G=gap.SmallGroup(128,960);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,512,422,1684,438,242,4037,1027,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^4*c^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^4*c^-1>;`
`// generators/relations`

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