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## G = D4.Q8order 64 = 26

### The non-split extension by D4 of Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for D4.Q8
G = < a,b,c,d | a4=b2=c4=1, d2=a2c2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Character table of D4.Q8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D size 1 1 1 1 4 4 2 2 2 2 4 4 4 8 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 2 2 0 0 2 -2 -2 2 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 -2 -2 -2 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 -2 2 0 -2 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ12 2 -2 2 -2 2 -2 0 -2 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 -2 2 -2 0 0 0 2 -2 0 2i 0 -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 0 0 0 2 -2 0 -2i 0 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 0 0 2i 0 0 -2i 0 0 0 0 0 √2 √-2 -√-2 -√2 complex lifted from C4○D8 ρ16 2 -2 -2 2 0 0 -2i 0 0 2i 0 0 0 0 0 √2 -√-2 √-2 -√2 complex lifted from C4○D8 ρ17 2 -2 -2 2 0 0 -2i 0 0 2i 0 0 0 0 0 -√2 √-2 -√-2 √2 complex lifted from C4○D8 ρ18 2 -2 -2 2 0 0 2i 0 0 -2i 0 0 0 0 0 -√2 -√-2 √-2 √2 complex lifted from C4○D8 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22

Smallest permutation representation of D4.Q8
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 6)(2 5)(3 8)(4 7)(9 16)(10 15)(11 14)(12 13)(17 23)(18 22)(19 21)(20 24)(25 31)(26 30)(27 29)(28 32)
(1 13 5 11)(2 14 6 12)(3 15 7 9)(4 16 8 10)(17 27 23 29)(18 28 24 30)(19 25 21 31)(20 26 22 32)
(1 21 7 17)(2 24 8 20)(3 23 5 19)(4 22 6 18)(9 31 13 27)(10 30 14 26)(11 29 15 25)(12 32 16 28)```

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

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

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

D4.Q8 is a maximal subgroup of
C42.D2p: C42.447D4  C42.219D4  C42.449D4  C42.384D4  C42.225D4  C42.229D4  C42.232D4  C42.280D4 ...
C4⋊C4.D2p: C42.20C23  C42.22C23  C42.23C23  C42.353C23  C42.354C23  C42.356C23  C42.357C23  C42.423C23 ...
D4.Q8 is a maximal quotient of
C4.Q89C4  C2.D85C4  (C2×C4).21Q16  C4.(C4⋊Q8)  (C2×C8).168D4  C4⋊C4.Q8
C4⋊C4.D2p: C2.(C4×D8)  D4⋊(C4⋊C4)  C42.31Q8  C4⋊C4.84D4  C2.(C8⋊Q8)  C4⋊C4.106D4  (C2×C4).23D8  D4.Dic6 ...
C42.D2p: C42.100D4  C42.123D4  D12.3Q8  D20.3Q8  D28.3Q8 ...

Matrix representation of D4.Q8 in GL4(𝔽17) generated by

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

D4.Q8 in GAP, Magma, Sage, TeX

`D_4.Q_8`
`% in TeX`

`G:=Group("D4.Q8");`
`// GroupNames label`

`G:=SmallGroup(64,159);`
`// by ID`

`G=gap.SmallGroup(64,159);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,247,362,1444,376,88]);`
`// Polycyclic`

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

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