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## G = Q8⋊2F7order 336 = 24·3·7

### The semidirect product of Q8 and F7 acting via F7/C7⋊C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — Q8⋊2F7
 Chief series C1 — C7 — C14 — C28 — C4×C7⋊C3 — C4⋊F7 — Q8⋊2F7
 Lower central C7 — C14 — C28 — Q8⋊2F7
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q82F7
G = < a,b,c,d | a4=c7=d6=1, b2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c5 >

Character table of Q82F7

 class 1 2A 2B 3A 3B 4A 4B 6A 6B 6C 6D 7 8A 8B 12A 12B 12C 12D 14 24A 24B 24C 24D 28A 28B 28C size 1 1 28 7 7 2 4 7 7 28 28 6 14 14 14 14 28 28 6 14 14 14 14 12 12 12 ρ1 1 1 1 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 -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 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 -1 1 -1 linear of order 2 ρ5 1 1 -1 ζ3 ζ32 1 1 ζ32 ζ3 ζ65 ζ6 1 -1 -1 ζ32 ζ3 ζ3 ζ32 1 ζ6 ζ65 ζ6 ζ65 1 1 1 linear of order 6 ρ6 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 1 1 linear of order 3 ρ7 1 1 1 ζ3 ζ32 1 -1 ζ32 ζ3 ζ3 ζ32 1 -1 -1 ζ32 ζ3 ζ65 ζ6 1 ζ6 ζ65 ζ6 ζ65 -1 1 -1 linear of order 6 ρ8 1 1 -1 ζ3 ζ32 1 -1 ζ32 ζ3 ζ65 ζ6 1 1 1 ζ32 ζ3 ζ65 ζ6 1 ζ32 ζ3 ζ32 ζ3 -1 1 -1 linear of order 6 ρ9 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 1 1 linear of order 3 ρ10 1 1 -1 ζ32 ζ3 1 1 ζ3 ζ32 ζ6 ζ65 1 -1 -1 ζ3 ζ32 ζ32 ζ3 1 ζ65 ζ6 ζ65 ζ6 1 1 1 linear of order 6 ρ11 1 1 1 ζ32 ζ3 1 -1 ζ3 ζ32 ζ32 ζ3 1 -1 -1 ζ3 ζ32 ζ6 ζ65 1 ζ65 ζ6 ζ65 ζ6 -1 1 -1 linear of order 6 ρ12 1 1 -1 ζ32 ζ3 1 -1 ζ3 ζ32 ζ6 ζ65 1 1 1 ζ3 ζ32 ζ6 ζ65 1 ζ3 ζ32 ζ3 ζ32 -1 1 -1 linear of order 6 ρ13 2 2 0 2 2 -2 0 2 2 0 0 2 0 0 -2 -2 0 0 2 0 0 0 0 0 -2 0 orthogonal lifted from D4 ρ14 2 2 0 -1+√-3 -1-√-3 -2 0 -1-√-3 -1+√-3 0 0 2 0 0 1+√-3 1-√-3 0 0 2 0 0 0 0 0 -2 0 complex lifted from C3×D4 ρ15 2 2 0 -1-√-3 -1+√-3 -2 0 -1+√-3 -1-√-3 0 0 2 0 0 1-√-3 1+√-3 0 0 2 0 0 0 0 0 -2 0 complex lifted from C3×D4 ρ16 2 -2 0 2 2 0 0 -2 -2 0 0 2 √-2 -√-2 0 0 0 0 -2 √-2 √-2 -√-2 -√-2 0 0 0 complex lifted from SD16 ρ17 2 -2 0 2 2 0 0 -2 -2 0 0 2 -√-2 √-2 0 0 0 0 -2 -√-2 -√-2 √-2 √-2 0 0 0 complex lifted from SD16 ρ18 2 -2 0 -1+√-3 -1-√-3 0 0 1+√-3 1-√-3 0 0 2 -√-2 √-2 0 0 0 0 -2 ζ87ζ32+ζ85ζ32 ζ87ζ3+ζ85ζ3 ζ83ζ32+ζ8ζ32 ζ83ζ3+ζ8ζ3 0 0 0 complex lifted from C3×SD16 ρ19 2 -2 0 -1-√-3 -1+√-3 0 0 1-√-3 1+√-3 0 0 2 √-2 -√-2 0 0 0 0 -2 ζ83ζ3+ζ8ζ3 ζ83ζ32+ζ8ζ32 ζ87ζ3+ζ85ζ3 ζ87ζ32+ζ85ζ32 0 0 0 complex lifted from C3×SD16 ρ20 2 -2 0 -1-√-3 -1+√-3 0 0 1-√-3 1+√-3 0 0 2 -√-2 √-2 0 0 0 0 -2 ζ87ζ3+ζ85ζ3 ζ87ζ32+ζ85ζ32 ζ83ζ3+ζ8ζ3 ζ83ζ32+ζ8ζ32 0 0 0 complex lifted from C3×SD16 ρ21 2 -2 0 -1+√-3 -1-√-3 0 0 1+√-3 1-√-3 0 0 2 √-2 -√-2 0 0 0 0 -2 ζ83ζ32+ζ8ζ32 ζ83ζ3+ζ8ζ3 ζ87ζ32+ζ85ζ32 ζ87ζ3+ζ85ζ3 0 0 0 complex lifted from C3×SD16 ρ22 6 6 0 0 0 6 -6 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 1 -1 1 orthogonal lifted from C2×F7 ρ23 6 6 0 0 0 6 6 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 orthogonal lifted from F7 ρ24 6 6 0 0 0 -6 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 -√-7 1 √-7 complex lifted from Dic7⋊C6 ρ25 6 6 0 0 0 -6 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 √-7 1 -√-7 complex lifted from Dic7⋊C6 ρ26 12 -12 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of Q82F7
On 56 points
Generators in S56
```(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)
(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)
(2 4 3 7 5 6)(9 11 10 14 12 13)(15 22)(16 25 17 28 19 27)(18 24 21 26 20 23)(29 43)(30 46 31 49 33 48)(32 45 35 47 34 44)(36 50)(37 53 38 56 40 55)(39 52 42 54 41 51)```

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

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

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

Matrix representation of Q82F7 in GL8(𝔽337)

 1 336 0 0 0 0 0 0 2 336 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 98 0 0 0 0 0 0 196 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 336 0 0 1 0 0 0 0 336 0 0 0 1 0 0 0 336 0 0 0 0 1 0 0 336 0 0 0 0 0 1 0 336 0 0 0 0 0 0 1 336
,
 128 209 0 0 0 0 0 0 0 209 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0

`G:=sub<GL(8,GF(337))| [1,2,0,0,0,0,0,0,336,336,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,196,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,336,336,336,336,336,336],[128,0,0,0,0,0,0,0,209,209,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;`

Q82F7 in GAP, Magma, Sage, TeX

`Q_8\rtimes_2F_7`
`% in TeX`

`G:=Group("Q8:2F7");`
`// GroupNames label`

`G:=SmallGroup(336,20);`
`// by ID`

`G=gap.SmallGroup(336,20);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,151,867,441,69,10373,1745]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^7=d^6=1,b^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=c^5>;`
`// generators/relations`

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