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## G = C8⋊Q8order 64 = 26

### The semidirect product of C8 and Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C8⋊Q8
G = < a,b,c | a8=b4=1, c2=b2, bab-1=a5, cac-1=a3, cbc-1=b-1 >

Character table of C8⋊Q8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 8 8 8 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 -2 0 2 symplectic lifted from Q8, Schur index 2 ρ12 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 2 0 -2 symplectic lifted from Q8, Schur index 2 ρ13 2 -2 2 -2 2 -2 0 0 0 0 0 0 2 0 -2 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 2 -2 2 -2 0 0 0 0 0 0 -2 0 2 0 symplectic lifted from Q8, Schur index 2 ρ15 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ16 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C8⋊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 33 22 28)(2 38 23 25)(3 35 24 30)(4 40 17 27)(5 37 18 32)(6 34 19 29)(7 39 20 26)(8 36 21 31)(9 44 60 54)(10 41 61 51)(11 46 62 56)(12 43 63 53)(13 48 64 50)(14 45 57 55)(15 42 58 52)(16 47 59 49)
(1 50 22 48)(2 53 23 43)(3 56 24 46)(4 51 17 41)(5 54 18 44)(6 49 19 47)(7 52 20 42)(8 55 21 45)(9 37 60 32)(10 40 61 27)(11 35 62 30)(12 38 63 25)(13 33 64 28)(14 36 57 31)(15 39 58 26)(16 34 59 29)```

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

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

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

Matrix representation of C8⋊Q8 in GL6(𝔽17)

 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 8 2 13 0 0 0 0 1 0 0 0 0 0 15 4 15
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 1 4 0 4 0 0 0 2 13 2 0 0 16 0 0 0
,
 13 0 0 0 0 0 0 4 0 0 0 0 0 0 13 14 16 13 0 0 9 5 13 16 0 0 11 9 15 15 0 0 16 14 13 1

`G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,2,1,15,0,0,1,13,0,4,0,0,0,0,0,15],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,16,0,0,0,4,2,0,0,0,0,0,13,0,0,0,1,4,2,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,9,11,16,0,0,14,5,9,14,0,0,16,13,15,13,0,0,13,16,15,1] >;`

C8⋊Q8 in GAP, Magma, Sage, TeX

`C_8\rtimes Q_8`
`% in TeX`

`G:=Group("C8:Q8");`
`// GroupNames label`

`G:=SmallGroup(64,182);`
`// by ID`

`G=gap.SmallGroup(64,182);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,55,362,332,86,1444,88]);`
`// Polycyclic`

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

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