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## G = Q8.F7order 336 = 24·3·7

### The non-split extension by Q8 of F7 acting via F7/D7=C3

Aliases: Q8.F7, Dic7.A4, C7⋊(C4.A4), C14.A41C2, C14.1(C2×A4), Q82D72C3, (C7×Q8).2C6, C2.2(D7⋊A4), SmallGroup(336,134)

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=1, b2=d6=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 3A 3B 4A 4B 4C 6A 6B 7 12A 12B 12C 12D 14 28A 28B 28C size 1 1 42 28 28 6 7 7 28 28 6 28 28 28 28 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 ζ3 ζ32 1 -1 -1 ζ3 ζ32 1 ζ6 ζ65 ζ65 ζ6 1 1 1 1 linear of order 6 ρ4 1 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 linear of order 3 ρ5 1 1 -1 ζ32 ζ3 1 -1 -1 ζ32 ζ3 1 ζ65 ζ6 ζ6 ζ65 1 1 1 1 linear of order 6 ρ6 1 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 linear of order 3 ρ7 2 -2 0 -1 -1 0 2i -2i 1 1 2 i -i i -i -2 0 0 0 complex lifted from C4.A4 ρ8 2 -2 0 -1 -1 0 -2i 2i 1 1 2 -i i -i i -2 0 0 0 complex lifted from C4.A4 ρ9 2 -2 0 ζ65 ζ6 0 -2i 2i ζ3 ζ32 2 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 -2 0 0 0 complex lifted from C4.A4 ρ10 2 -2 0 ζ6 ζ65 0 -2i 2i ζ32 ζ3 2 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 -2 0 0 0 complex lifted from C4.A4 ρ11 2 -2 0 ζ65 ζ6 0 2i -2i ζ3 ζ32 2 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 -2 0 0 0 complex lifted from C4.A4 ρ12 2 -2 0 ζ6 ζ65 0 2i -2i ζ32 ζ3 2 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 -2 0 0 0 complex lifted from C4.A4 ρ13 3 3 -1 0 0 -1 3 3 0 0 3 0 0 0 0 3 -1 -1 -1 orthogonal lifted from A4 ρ14 3 3 1 0 0 -1 -3 -3 0 0 3 0 0 0 0 3 -1 -1 -1 orthogonal lifted from C2×A4 ρ15 6 6 0 0 0 6 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F7 ρ16 6 6 0 0 0 -2 0 0 0 0 -1 0 0 0 0 -1 2ζ75+2ζ72+1 2ζ76+2ζ7+1 2ζ74+2ζ73+1 orthogonal lifted from D7⋊A4 ρ17 6 6 0 0 0 -2 0 0 0 0 -1 0 0 0 0 -1 2ζ76+2ζ7+1 2ζ74+2ζ73+1 2ζ75+2ζ72+1 orthogonal lifted from D7⋊A4 ρ18 6 6 0 0 0 -2 0 0 0 0 -1 0 0 0 0 -1 2ζ74+2ζ73+1 2ζ75+2ζ72+1 2ζ76+2ζ7+1 orthogonal lifted from D7⋊A4 ρ19 12 -12 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of Q8.F7
On 112 points
Generators in S112
```(1 12 3 6)(2 9 4 15)(5 7 11 13)(8 10 14 16)(17 108 23 102)(18 65 24 71)(19 57 25 63)(20 111 26 105)(21 68 27 74)(22 60 28 54)(29 51 35 45)(30 96 36 90)(31 77 37 83)(32 42 38 48)(33 99 39 93)(34 80 40 86)(41 91 47 97)(43 79 49 85)(44 94 50 100)(46 82 52 88)(53 112 59 106)(55 76 61 70)(56 103 62 109)(58 67 64 73)(66 110 72 104)(69 101 75 107)(78 98 84 92)(81 89 87 95)
(1 8 3 14)(2 5 4 11)(6 10 12 16)(7 9 13 15)(17 76 23 70)(18 56 24 62)(19 110 25 104)(20 67 26 73)(21 59 27 53)(22 101 28 107)(29 95 35 89)(30 88 36 82)(31 41 37 47)(32 98 38 92)(33 79 39 85)(34 44 40 50)(42 78 48 84)(43 93 49 99)(45 81 51 87)(46 96 52 90)(54 75 60 69)(55 102 61 108)(57 66 63 72)(58 105 64 111)(65 109 71 103)(68 112 74 106)(77 97 83 91)(80 100 86 94)
(1 81 77 106 85 110 102)(2 111 107 82 103 86 78)(3 87 83 112 79 104 108)(4 105 101 88 109 80 84)(5 58 22 30 65 94 48)(6 95 31 59 49 66 23)(7 67 60 96 24 50 32)(8 51 97 68 33 25 61)(9 26 69 52 62 34 98)(10 35 41 27 99 63 70)(11 64 28 36 71 100 42)(12 89 37 53 43 72 17)(13 73 54 90 18 44 38)(14 45 91 74 39 19 55)(15 20 75 46 56 40 92)(16 29 47 21 93 57 76)
(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)(65 66 67 68 69 70 71 72 73 74 75 76)(77 78 79 80 81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112)```

`G:=sub<Sym(112)| (1,12,3,6)(2,9,4,15)(5,7,11,13)(8,10,14,16)(17,108,23,102)(18,65,24,71)(19,57,25,63)(20,111,26,105)(21,68,27,74)(22,60,28,54)(29,51,35,45)(30,96,36,90)(31,77,37,83)(32,42,38,48)(33,99,39,93)(34,80,40,86)(41,91,47,97)(43,79,49,85)(44,94,50,100)(46,82,52,88)(53,112,59,106)(55,76,61,70)(56,103,62,109)(58,67,64,73)(66,110,72,104)(69,101,75,107)(78,98,84,92)(81,89,87,95), (1,8,3,14)(2,5,4,11)(6,10,12,16)(7,9,13,15)(17,76,23,70)(18,56,24,62)(19,110,25,104)(20,67,26,73)(21,59,27,53)(22,101,28,107)(29,95,35,89)(30,88,36,82)(31,41,37,47)(32,98,38,92)(33,79,39,85)(34,44,40,50)(42,78,48,84)(43,93,49,99)(45,81,51,87)(46,96,52,90)(54,75,60,69)(55,102,61,108)(57,66,63,72)(58,105,64,111)(65,109,71,103)(68,112,74,106)(77,97,83,91)(80,100,86,94), (1,81,77,106,85,110,102)(2,111,107,82,103,86,78)(3,87,83,112,79,104,108)(4,105,101,88,109,80,84)(5,58,22,30,65,94,48)(6,95,31,59,49,66,23)(7,67,60,96,24,50,32)(8,51,97,68,33,25,61)(9,26,69,52,62,34,98)(10,35,41,27,99,63,70)(11,64,28,36,71,100,42)(12,89,37,53,43,72,17)(13,73,54,90,18,44,38)(14,45,91,74,39,19,55)(15,20,75,46,56,40,92)(16,29,47,21,93,57,76), (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)(65,66,67,68,69,70,71,72,73,74,75,76)(77,78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112)>;`

`G:=Group( (1,12,3,6)(2,9,4,15)(5,7,11,13)(8,10,14,16)(17,108,23,102)(18,65,24,71)(19,57,25,63)(20,111,26,105)(21,68,27,74)(22,60,28,54)(29,51,35,45)(30,96,36,90)(31,77,37,83)(32,42,38,48)(33,99,39,93)(34,80,40,86)(41,91,47,97)(43,79,49,85)(44,94,50,100)(46,82,52,88)(53,112,59,106)(55,76,61,70)(56,103,62,109)(58,67,64,73)(66,110,72,104)(69,101,75,107)(78,98,84,92)(81,89,87,95), (1,8,3,14)(2,5,4,11)(6,10,12,16)(7,9,13,15)(17,76,23,70)(18,56,24,62)(19,110,25,104)(20,67,26,73)(21,59,27,53)(22,101,28,107)(29,95,35,89)(30,88,36,82)(31,41,37,47)(32,98,38,92)(33,79,39,85)(34,44,40,50)(42,78,48,84)(43,93,49,99)(45,81,51,87)(46,96,52,90)(54,75,60,69)(55,102,61,108)(57,66,63,72)(58,105,64,111)(65,109,71,103)(68,112,74,106)(77,97,83,91)(80,100,86,94), (1,81,77,106,85,110,102)(2,111,107,82,103,86,78)(3,87,83,112,79,104,108)(4,105,101,88,109,80,84)(5,58,22,30,65,94,48)(6,95,31,59,49,66,23)(7,67,60,96,24,50,32)(8,51,97,68,33,25,61)(9,26,69,52,62,34,98)(10,35,41,27,99,63,70)(11,64,28,36,71,100,42)(12,89,37,53,43,72,17)(13,73,54,90,18,44,38)(14,45,91,74,39,19,55)(15,20,75,46,56,40,92)(16,29,47,21,93,57,76), (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)(65,66,67,68,69,70,71,72,73,74,75,76)(77,78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112) );`

`G=PermutationGroup([[(1,12,3,6),(2,9,4,15),(5,7,11,13),(8,10,14,16),(17,108,23,102),(18,65,24,71),(19,57,25,63),(20,111,26,105),(21,68,27,74),(22,60,28,54),(29,51,35,45),(30,96,36,90),(31,77,37,83),(32,42,38,48),(33,99,39,93),(34,80,40,86),(41,91,47,97),(43,79,49,85),(44,94,50,100),(46,82,52,88),(53,112,59,106),(55,76,61,70),(56,103,62,109),(58,67,64,73),(66,110,72,104),(69,101,75,107),(78,98,84,92),(81,89,87,95)], [(1,8,3,14),(2,5,4,11),(6,10,12,16),(7,9,13,15),(17,76,23,70),(18,56,24,62),(19,110,25,104),(20,67,26,73),(21,59,27,53),(22,101,28,107),(29,95,35,89),(30,88,36,82),(31,41,37,47),(32,98,38,92),(33,79,39,85),(34,44,40,50),(42,78,48,84),(43,93,49,99),(45,81,51,87),(46,96,52,90),(54,75,60,69),(55,102,61,108),(57,66,63,72),(58,105,64,111),(65,109,71,103),(68,112,74,106),(77,97,83,91),(80,100,86,94)], [(1,81,77,106,85,110,102),(2,111,107,82,103,86,78),(3,87,83,112,79,104,108),(4,105,101,88,109,80,84),(5,58,22,30,65,94,48),(6,95,31,59,49,66,23),(7,67,60,96,24,50,32),(8,51,97,68,33,25,61),(9,26,69,52,62,34,98),(10,35,41,27,99,63,70),(11,64,28,36,71,100,42),(12,89,37,53,43,72,17),(13,73,54,90,18,44,38),(14,45,91,74,39,19,55),(15,20,75,46,56,40,92),(16,29,47,21,93,57,76)], [(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),(65,66,67,68,69,70,71,72,73,74,75,76),(77,78,79,80,81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112)]])`

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 55 0 24 142 142 24 0 0 313 31 313 0 118 118 0 0 219 195 250 195 219 0 0 0 0 219 195 250 195 219 0 0 118 118 0 313 31 313 0 0 24 142 142 24 0 55
,
 128 129 0 0 0 0 0 0 129 209 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
,
 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
,
 189 0 0 0 0 0 0 0 72 220 0 0 0 0 0 0 0 0 31 17 17 0 17 0 0 0 0 320 0 320 320 14 0 0 17 0 0 31 17 17 0 0 0 31 17 17 0 17 0 0 323 323 306 323 306 306 0 0 320 0 320 320 14 0

`G:=sub<GL(8,GF(337))| [0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,55,313,219,0,118,24,0,0,0,31,195,219,118,142,0,0,24,313,250,195,0,142,0,0,142,0,195,250,313,24,0,0,142,118,219,195,31,0,0,0,24,118,0,219,313,55],[128,129,0,0,0,0,0,0,129,209,0,0,0,0,0,0,0,0,250,142,313,0,24,195,0,0,0,55,118,313,24,219,0,0,195,142,31,118,0,219,0,0,219,0,118,31,142,195,0,0,219,24,313,118,55,0,0,0,195,24,0,313,142,250],[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],[189,72,0,0,0,0,0,0,0,220,0,0,0,0,0,0,0,0,31,0,17,0,323,320,0,0,17,320,0,31,323,0,0,0,17,0,0,17,306,320,0,0,0,320,31,17,323,320,0,0,17,320,17,0,306,14,0,0,0,14,17,17,306,0] >;`

Q8.F7 in GAP, Magma, Sage, TeX

`Q_8.F_7`
`% in TeX`

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

`G:=SmallGroup(336,134);`
`// by ID`

`G=gap.SmallGroup(336,134);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,2,-7,-2,1008,116,518,225,735,357,4324,1450]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^7=1,b^2=d^6=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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