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## G = Q8⋊F7order 336 = 24·3·7

### The semidirect product of Q8 and F7 acting via F7/D7=C3

Aliases: Q8⋊F7, D14.A4, D7⋊SL2(𝔽3), (Q8×D7)⋊2C3, (C7×Q8)⋊2C6, C14.A42C2, C14.2(C2×A4), C7⋊(C2×SL2(𝔽3)), C2.3(D7⋊A4), SmallGroup(336,135)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×Q8 — Q8⋊F7
 Chief series C1 — C2 — C14 — C7×Q8 — C14.A4 — Q8⋊F7
 Lower central C7×Q8 — Q8⋊F7
 Upper central C1 — C2

Generators and relations for Q8⋊F7
G = < a,b,c,d | a4=c7=d6=1, b2=a2, bab-1=a-1, ac=ca, dad-1=b, bc=cb, dbd-1=ab, dcd-1=c5 >

Character table of Q8⋊F7

 class 1 2A 2B 2C 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 7 14 28A 28B 28C size 1 1 7 7 28 28 6 42 28 28 28 28 28 28 6 6 12 12 12 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 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 linear of order 3 ρ4 1 1 -1 -1 ζ32 ζ3 1 -1 ζ6 ζ3 ζ65 ζ65 ζ6 ζ32 1 1 1 1 1 linear of order 6 ρ5 1 1 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 linear of order 3 ρ6 1 1 -1 -1 ζ3 ζ32 1 -1 ζ65 ζ32 ζ6 ζ6 ζ65 ζ3 1 1 1 1 1 linear of order 6 ρ7 2 -2 2 -2 -1 -1 0 0 -1 1 1 -1 1 1 2 -2 0 0 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ8 2 -2 -2 2 -1 -1 0 0 1 1 -1 1 -1 1 2 -2 0 0 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ9 2 -2 -2 2 ζ65 ζ6 0 0 ζ3 ζ32 ζ6 ζ32 ζ65 ζ3 2 -2 0 0 0 complex lifted from SL2(𝔽3) ρ10 2 -2 -2 2 ζ6 ζ65 0 0 ζ32 ζ3 ζ65 ζ3 ζ6 ζ32 2 -2 0 0 0 complex lifted from SL2(𝔽3) ρ11 2 -2 2 -2 ζ6 ζ65 0 0 ζ6 ζ3 ζ3 ζ65 ζ32 ζ32 2 -2 0 0 0 complex lifted from SL2(𝔽3) ρ12 2 -2 2 -2 ζ65 ζ6 0 0 ζ65 ζ32 ζ32 ζ6 ζ3 ζ3 2 -2 0 0 0 complex lifted from SL2(𝔽3) ρ13 3 3 -3 -3 0 0 -1 1 0 0 0 0 0 0 3 3 -1 -1 -1 orthogonal lifted from C2×A4 ρ14 3 3 3 3 0 0 -1 -1 0 0 0 0 0 0 3 3 -1 -1 -1 orthogonal lifted from A4 ρ15 6 6 0 0 0 0 6 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from F7 ρ16 6 6 0 0 0 0 -2 0 0 0 0 0 0 0 -1 -1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 orthogonal lifted from D7⋊A4 ρ17 6 6 0 0 0 0 -2 0 0 0 0 0 0 0 -1 -1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 orthogonal lifted from D7⋊A4 ρ18 6 6 0 0 0 0 -2 0 0 0 0 0 0 0 -1 -1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 orthogonal lifted from D7⋊A4 ρ19 12 -12 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Q8⋊F7
On 56 points
Generators in S56
```(1 25 17 31)(2 26 18 32)(3 27 19 33)(4 28 20 34)(5 22 21 35)(6 23 15 29)(7 24 16 30)(8 43 51 37)(9 44 52 38)(10 45 53 39)(11 46 54 40)(12 47 55 41)(13 48 56 42)(14 49 50 36)
(1 39 17 45)(2 40 18 46)(3 41 19 47)(4 42 20 48)(5 36 21 49)(6 37 15 43)(7 38 16 44)(8 23 51 29)(9 24 52 30)(10 25 53 31)(11 26 54 32)(12 27 55 33)(13 28 56 34)(14 22 50 35)
(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)(8 46 34 12 44 35)(9 49 29 11 48 33)(10 45 31)(13 47 30 14 43 32)(15 18 20 19 16 21)(22 51 40 28 55 38)(23 54 42 27 52 36)(24 50 37 26 56 41)(25 53 39)```

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

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

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

Matrix representation of Q8⋊F7 in GL8(𝔽337)

 0 336 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 250 0 195 219 219 195 0 0 142 55 142 0 24 24 0 0 313 118 31 118 313 0 0 0 0 313 118 31 118 313 0 0 24 24 0 142 55 142 0 0 195 219 219 195 0 250
,
 208 209 0 0 0 0 0 0 209 129 0 0 0 0 0 0 0 0 31 0 118 313 313 118 0 0 219 250 219 0 195 195 0 0 142 24 55 24 142 0 0 0 0 142 24 55 24 142 0 0 195 195 0 219 250 219 0 0 118 313 313 118 0 31
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 336 336 336 336 336 336 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
,
 336 0 0 0 0 0 0 0 208 209 0 0 0 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 0 0 336 336 336 336 336 336 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(337))| [0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,250,142,313,0,24,195,0,0,0,55,118,313,24,219,0,0,195,142,31,118,0,219,0,0,219,0,118,31,142,195,0,0,219,24,313,118,55,0,0,0,195,24,0,313,142,250],[208,209,0,0,0,0,0,0,209,129,0,0,0,0,0,0,0,0,31,219,142,0,195,118,0,0,0,250,24,142,195,313,0,0,118,219,55,24,0,313,0,0,313,0,24,55,219,118,0,0,313,195,142,24,250,0,0,0,118,195,0,142,219,31],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0,1,0,0,0,0,0,336,0,0,1,0,0,0,0,336,0,0,0,1,0,0,0,336,0,0,0,0,1,0,0,336,0,0,0,0,0],[336,208,0,0,0,0,0,0,0,209,0,0,0,0,0,0,0,0,1,0,0,0,336,0,0,0,0,0,0,1,336,0,0,0,0,0,0,0,336,0,0,0,0,0,1,0,336,0,0,0,0,0,0,0,336,1,0,0,0,1,0,0,336,0] >;`

Q8⋊F7 in GAP, Magma, Sage, TeX

`Q_8\rtimes F_7`
`% in TeX`

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

`G:=SmallGroup(336,135);`
`// by ID`

`G=gap.SmallGroup(336,135);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,2,-7,-2,116,518,225,735,357,4324,1450]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^7=d^6=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=b,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^5>;`
`// generators/relations`

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