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## G = Q8.D4order 64 = 26

### 2nd non-split extension by Q8 of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8.D4
G = < a,b,c,d | a4=c4=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >

Character table of Q8.D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D size 1 1 1 1 8 2 2 2 2 4 4 4 4 4 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 2 -2 0 0 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 0 2 -2 0 0 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 -2 -2 2 2 0 0 0 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 2 -2 0 -2 2 0 0 2i -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 0 -2 2 0 0 -2i 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 0 0 0 -2i 2i 0 0 0 0 0 0 -√-2 √-2 √2 -√2 complex lifted from C4○D8 ρ16 2 -2 -2 2 0 0 0 -2i 2i 0 0 0 0 0 0 √-2 -√-2 -√2 √2 complex lifted from C4○D8 ρ17 2 -2 -2 2 0 0 0 2i -2i 0 0 0 0 0 0 -√-2 √-2 -√2 √2 complex lifted from C4○D8 ρ18 2 -2 -2 2 0 0 0 2i -2i 0 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 symplectic lifted from C8.C22, Schur index 2

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

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

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

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

Q8.D4 is a maximal subgroup of
Q8.2D12
C42.D2p: C42.443D4  C42.212D4  C42.446D4  C42.384D4  C42.226D4  C42.229D4  C42.235D4  C42.268D4 ...
(Cp×Q8).D4: C42.17C23  C42.505C23  C42.506C23  C42.510C23  C42.512C23  C42.516C23  C42.518C23  C42.527C23 ...
C4⋊C4.D2p: C42.16C23  C42.18C23  C42.19C23  C42.355C23  C42.358C23  C42.359C23  C42.408C23  C42.409C23 ...
Q8.D4 is a maximal quotient of
D4⋊C4⋊C4  C4.68(C4×D4)  C42.31Q8  (C2×C8).24Q8
(Cp×Q8).D4: Q8.2SD16  Q8.3SD16  Q8.2D8  Q8.2Q16  C42.249C23  C42.251C23  C42.253C23  C42.255C23 ...
C42.D2p: C42.119D4  C42.36D6  C42.61D6  C42.36D10  C42.61D10  C42.36D14  C42.61D14 ...
(C2×C8).D2p: (C2×Q8).8Q8  (C2×C4).23D8  Dic6.D4  Dic6.11D4  Dic10.D4  Dic10.11D4  Dic14.D4  Dic14.11D4 ...

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

 1 0 0 0 0 1 0 0 0 0 1 15 0 0 1 16
,
 16 0 0 0 0 16 0 0 0 0 10 7 0 0 5 7
,
 16 16 0 0 2 1 0 0 0 0 4 0 0 0 0 4
,
 16 16 0 0 0 1 0 0 0 0 4 0 0 0 4 13
`G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[16,0,0,0,0,16,0,0,0,0,10,5,0,0,7,7],[16,2,0,0,16,1,0,0,0,0,4,0,0,0,0,4],[16,0,0,0,16,1,0,0,0,0,4,4,0,0,0,13] >;`

Q8.D4 in GAP, Magma, Sage, TeX

`Q_8.D_4`
`% in TeX`

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

`G:=SmallGroup(64,145);`
`// by ID`

`G=gap.SmallGroup(64,145);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,158,1444,376,88]);`
`// Polycyclic`

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

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