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## G = Q8⋊Q8order 64 = 26

### 1st semidirect product of Q8 and 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 — Q8⋊Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×Q8 — C4×Q8 — Q8⋊Q8
 Lower central C1 — C2 — C2×C4 — Q8⋊Q8
 Upper central C1 — C22 — C42 — Q8⋊Q8
 Jennings C1 — C2 — C2 — C2×C4 — Q8⋊Q8

Generators and relations for Q8⋊Q8
G = < a,b,c,d | a4=c4=1, b2=a2, d2=c2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >

Character table of Q8⋊Q8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D size 1 1 1 1 2 2 2 2 4 4 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 2 -2 -2 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 -2 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 2 -2 0 0 0 0 2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ12 2 -2 2 -2 0 2 -2 0 0 0 0 -2 2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 -2 2 -2 0 -2 2 0 2i 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 0 -2 2 0 -2i 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 -2 0 0 2 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ16 2 -2 -2 2 2 0 0 -2 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ17 2 -2 -2 2 2 0 0 -2 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ18 2 -2 -2 2 -2 0 0 2 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 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⋊Q8
Regular action 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 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)(33 44 35 42)(34 43 36 41)(37 48 39 46)(38 47 40 45)(49 60 51 58)(50 59 52 57)(53 64 55 62)(54 63 56 61)
(1 23 7 19)(2 24 8 20)(3 21 5 17)(4 22 6 18)(9 29 13 25)(10 30 14 26)(11 31 15 27)(12 32 16 28)(33 49 37 53)(34 50 38 54)(35 51 39 55)(36 52 40 56)(41 57 45 61)(42 58 46 62)(43 59 47 63)(44 60 48 64)
(1 37 7 33)(2 40 8 36)(3 39 5 35)(4 38 6 34)(9 47 13 43)(10 46 14 42)(11 45 15 41)(12 48 16 44)(17 55 21 51)(18 54 22 50)(19 53 23 49)(20 56 24 52)(25 63 29 59)(26 62 30 58)(27 61 31 57)(28 64 32 60)```

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

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

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

Matrix representation of Q8⋊Q8 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 4 0 0 0 0 13
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 16 0
,
 0 4 0 0 4 0 0 0 0 0 1 0 0 0 0 1
,
 10 13 0 0 4 7 0 0 0 0 0 9 0 0 2 0
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[0,1,0,0,1,0,0,0,0,0,0,16,0,0,1,0],[0,4,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[10,4,0,0,13,7,0,0,0,0,0,2,0,0,9,0] >;`

Q8⋊Q8 in GAP, Magma, Sage, TeX

`Q_8\rtimes Q_8`
`% in TeX`

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

`G:=SmallGroup(64,156);`
`// by ID`

`G=gap.SmallGroup(64,156);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,55,362,158,1444,376,88]);`
`// Polycyclic`

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

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