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## G = D8.2Q8order 128 = 27

### 2nd 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 — D8.2Q8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C4○D8 — C8○D8 — D8.2Q8
 Lower central C1 — C2 — C4 — C2×C8 — D8.2Q8
 Upper central C1 — C2 — C2×C4 — C4×C8 — D8.2Q8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8.2Q8

Generators and relations for D8.2Q8
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6c2, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a5b, dcd-1=a4c3 >

Subgroups: 132 in 60 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×4], C22, C22, C8 [×4], C8, C2×C4, C2×C4 [×4], D4 [×2], Q8, C16 [×2], C42, C4⋊C4 [×4], C2×C8 [×2], C2×C8, M4(2) [×2], D8, SD16, Q16, C4○D4, C4×C8, C4≀C2, C4.Q8 [×2], C2.D8, C8.C4, M5(2) [×2], C42.C2, C8○D4, C4○D8, D82C4 [×2], C8.C8, C8.Q8 [×2], C8○D8, C8.5Q8, D8.2Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, SD16 [×2], C2×D4, C2×Q8, C4○D4, C22⋊Q8, C2×SD16, C8⋊C22, D42Q8, D8.2Q8

Character table of D8.2Q8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H 16A 16B 16C 16D size 1 1 2 8 2 2 4 4 8 16 16 2 2 2 2 4 4 8 8 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 0 2 2 -2 -2 0 0 0 -2 -2 2 2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 2 2 2 2 0 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 -2 2 0 0 -2 0 0 -2 -2 0 0 2 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ12 2 2 -2 -2 -2 2 0 0 2 0 0 -2 -2 0 0 2 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 2 -2 0 -2 2 0 0 0 0 0 2 2 0 0 -2 0 2i -2i 0 0 0 0 complex lifted from C4○D4 ρ14 2 2 -2 0 -2 2 0 0 0 0 0 2 2 0 0 -2 0 -2i 2i 0 0 0 0 complex lifted from C4○D4 ρ15 2 2 -2 0 2 -2 0 0 0 0 0 0 0 -2 -2 0 2 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ16 2 2 -2 0 2 -2 0 0 0 0 0 0 0 2 2 0 -2 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ17 2 2 -2 0 2 -2 0 0 0 0 0 0 0 2 2 0 -2 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ18 2 2 -2 0 2 -2 0 0 0 0 0 0 0 -2 -2 0 2 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ19 4 4 4 0 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 0 0 0 0 2i -2i 0 0 0 -2√-2 2√-2 2√2 -2√2 0 0 0 0 0 0 0 0 complex faithful ρ21 4 -4 0 0 0 0 -2i 2i 0 0 0 -2√-2 2√-2 -2√2 2√2 0 0 0 0 0 0 0 0 complex faithful ρ22 4 -4 0 0 0 0 2i -2i 0 0 0 2√-2 -2√-2 -2√2 2√2 0 0 0 0 0 0 0 0 complex faithful ρ23 4 -4 0 0 0 0 -2i 2i 0 0 0 2√-2 -2√-2 2√2 -2√2 0 0 0 0 0 0 0 0 complex faithful

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

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

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

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

Matrix representation of D8.2Q8 in GL4(𝔽17) generated by

 5 12 0 0 5 5 0 0 0 0 12 12 0 0 5 12
,
 0 0 12 12 0 0 5 12 5 12 0 0 5 5 0 0
,
 14 14 0 0 3 14 0 0 0 0 14 14 0 0 3 14
,
 14 14 0 0 14 3 0 0 0 0 13 0 0 0 0 4
`G:=sub<GL(4,GF(17))| [5,5,0,0,12,5,0,0,0,0,12,5,0,0,12,12],[0,0,5,5,0,0,12,5,12,5,0,0,12,12,0,0],[14,3,0,0,14,14,0,0,0,0,14,3,0,0,14,14],[14,14,0,0,14,3,0,0,0,0,13,0,0,0,0,4] >;`

D8.2Q8 in GAP, Magma, Sage, TeX

`D_8._2Q_8`
`% in TeX`

`G:=Group("D8.2Q8");`
`// GroupNames label`

`G:=SmallGroup(128,963);`
`// by ID`

`G=gap.SmallGroup(128,963);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,280,141,512,422,2019,248,1684,998,102,4037,1027,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^4*c^3>;`
`// generators/relations`

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