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## G = Q8.2F7order 336 = 24·3·7

### The non-split extension by Q8 of 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 Q8.2F7
G = < a,b,c,d | a4=c7=1, b2=d6=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c5 >

Character table of Q8.2F7

 class 1 2 3A 3B 4A 4B 4C 6A 6B 7 8A 8B 12A 12B 12C 12D 12E 12F 14 24A 24B 24C 24D 28A 28B 28C size 1 1 7 7 2 4 28 7 7 6 14 14 14 14 28 28 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 ζ32 ζ3 1 -1 1 ζ3 ζ32 1 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ65 ζ6 1 ζ65 ζ6 ζ65 ζ6 -1 1 -1 linear of order 6 ρ6 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 1 1 1 linear of order 3 ρ7 1 1 ζ3 ζ32 1 -1 -1 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 1 ζ32 ζ3 ζ32 ζ3 -1 1 -1 linear of order 6 ρ8 1 1 ζ32 ζ3 1 1 -1 ζ3 ζ32 1 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ3 ζ32 1 ζ65 ζ6 ζ65 ζ6 1 1 1 linear of order 6 ρ9 1 1 ζ3 ζ32 1 -1 1 ζ32 ζ3 1 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ6 ζ65 1 ζ6 ζ65 ζ6 ζ65 -1 1 -1 linear of order 6 ρ10 1 1 ζ3 ζ32 1 1 -1 ζ32 ζ3 1 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ32 ζ3 1 ζ6 ζ65 ζ6 ζ65 1 1 1 linear of order 6 ρ11 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 1 1 1 linear of order 3 ρ12 1 1 ζ32 ζ3 1 -1 -1 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 1 ζ3 ζ32 ζ3 ζ32 -1 1 -1 linear of order 6 ρ13 2 2 2 2 -2 0 0 2 2 2 0 0 -2 -2 0 0 0 0 2 0 0 0 0 0 -2 0 orthogonal lifted from D4 ρ14 2 -2 2 2 0 0 0 -2 -2 2 √2 -√2 0 0 0 0 0 0 -2 √2 √2 -√2 -√2 0 0 0 symplectic lifted from Q16, Schur index 2 ρ15 2 -2 2 2 0 0 0 -2 -2 2 -√2 √2 0 0 0 0 0 0 -2 -√2 -√2 √2 √2 0 0 0 symplectic lifted from Q16, Schur index 2 ρ16 2 2 -1-√-3 -1+√-3 -2 0 0 -1+√-3 -1-√-3 2 0 0 1+√-3 1-√-3 0 0 0 0 2 0 0 0 0 0 -2 0 complex lifted from C3×D4 ρ17 2 2 -1+√-3 -1-√-3 -2 0 0 -1-√-3 -1+√-3 2 0 0 1-√-3 1+√-3 0 0 0 0 2 0 0 0 0 0 -2 0 complex lifted from C3×D4 ρ18 2 -2 -1-√-3 -1+√-3 0 0 0 1-√-3 1+√-3 2 -√2 √2 0 0 0 0 0 0 -2 ζ83ζ3-ζ8ζ3 ζ83ζ32-ζ8ζ32 ζ87ζ3-ζ85ζ3 ζ87ζ32-ζ85ζ32 0 0 0 complex lifted from C3×Q16 ρ19 2 -2 -1+√-3 -1-√-3 0 0 0 1+√-3 1-√-3 2 -√2 √2 0 0 0 0 0 0 -2 ζ83ζ32-ζ8ζ32 ζ83ζ3-ζ8ζ3 ζ87ζ32-ζ85ζ32 ζ87ζ3-ζ85ζ3 0 0 0 complex lifted from C3×Q16 ρ20 2 -2 -1-√-3 -1+√-3 0 0 0 1-√-3 1+√-3 2 √2 -√2 0 0 0 0 0 0 -2 ζ87ζ3-ζ85ζ3 ζ87ζ32-ζ85ζ32 ζ83ζ3-ζ8ζ3 ζ83ζ32-ζ8ζ32 0 0 0 complex lifted from C3×Q16 ρ21 2 -2 -1+√-3 -1-√-3 0 0 0 1+√-3 1-√-3 2 √2 -√2 0 0 0 0 0 0 -2 ζ87ζ32-ζ85ζ32 ζ87ζ3-ζ85ζ3 ζ83ζ32-ζ8ζ32 ζ83ζ3-ζ8ζ3 0 0 0 complex lifted from C3×Q16 ρ22 6 6 0 0 6 -6 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 1 orthogonal lifted from C2×F7 ρ23 6 6 0 0 6 6 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 orthogonal lifted from F7 ρ24 6 6 0 0 -6 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 √-7 1 -√-7 complex lifted from Dic7⋊C6 ρ25 6 6 0 0 -6 0 0 0 0 -1 0 0 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 -2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Q8.2F7
On 112 points
Generators in S112
```(1 7 3 5)(2 6 4 8)(9 13 11 15)(10 16 12 14)(17 45 23 51)(18 52 24 46)(19 47 25 41)(20 42 26 48)(21 49 27 43)(22 44 28 50)(29 80 35 86)(30 87 36 81)(31 82 37 88)(32 77 38 83)(33 84 39 78)(34 79 40 85)(53 93 59 99)(54 100 60 94)(55 95 61 89)(56 90 62 96)(57 97 63 91)(58 92 64 98)(65 112 71 106)(66 107 72 101)(67 102 73 108)(68 109 74 103)(69 104 75 110)(70 111 76 105)
(1 14 3 16)(2 11 4 9)(5 10 7 12)(6 13 8 15)(17 110 23 104)(18 70 24 76)(19 112 25 106)(20 72 26 66)(21 102 27 108)(22 74 28 68)(29 61 35 55)(30 90 36 96)(31 63 37 57)(32 92 38 98)(33 53 39 59)(34 94 40 100)(41 71 47 65)(42 107 48 101)(43 73 49 67)(44 109 50 103)(45 75 51 69)(46 111 52 105)(54 79 60 85)(56 81 62 87)(58 83 64 77)(78 93 84 99)(80 95 86 89)(82 97 88 91)
(1 23 19 63 27 55 59)(2 56 64 24 60 28 20)(3 17 25 57 21 61 53)(4 62 58 18 54 22 26)(5 45 41 97 49 89 93)(6 90 98 46 94 50 42)(7 51 47 91 43 95 99)(8 96 92 52 100 44 48)(9 87 83 70 79 74 66)(10 75 71 88 67 80 84)(11 81 77 76 85 68 72)(12 69 65 82 73 86 78)(13 36 32 111 40 103 107)(14 104 112 37 108 29 33)(15 30 38 105 34 109 101)(16 110 106 31 102 35 39)
(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)| (1,7,3,5)(2,6,4,8)(9,13,11,15)(10,16,12,14)(17,45,23,51)(18,52,24,46)(19,47,25,41)(20,42,26,48)(21,49,27,43)(22,44,28,50)(29,80,35,86)(30,87,36,81)(31,82,37,88)(32,77,38,83)(33,84,39,78)(34,79,40,85)(53,93,59,99)(54,100,60,94)(55,95,61,89)(56,90,62,96)(57,97,63,91)(58,92,64,98)(65,112,71,106)(66,107,72,101)(67,102,73,108)(68,109,74,103)(69,104,75,110)(70,111,76,105), (1,14,3,16)(2,11,4,9)(5,10,7,12)(6,13,8,15)(17,110,23,104)(18,70,24,76)(19,112,25,106)(20,72,26,66)(21,102,27,108)(22,74,28,68)(29,61,35,55)(30,90,36,96)(31,63,37,57)(32,92,38,98)(33,53,39,59)(34,94,40,100)(41,71,47,65)(42,107,48,101)(43,73,49,67)(44,109,50,103)(45,75,51,69)(46,111,52,105)(54,79,60,85)(56,81,62,87)(58,83,64,77)(78,93,84,99)(80,95,86,89)(82,97,88,91), (1,23,19,63,27,55,59)(2,56,64,24,60,28,20)(3,17,25,57,21,61,53)(4,62,58,18,54,22,26)(5,45,41,97,49,89,93)(6,90,98,46,94,50,42)(7,51,47,91,43,95,99)(8,96,92,52,100,44,48)(9,87,83,70,79,74,66)(10,75,71,88,67,80,84)(11,81,77,76,85,68,72)(12,69,65,82,73,86,78)(13,36,32,111,40,103,107)(14,104,112,37,108,29,33)(15,30,38,105,34,109,101)(16,110,106,31,102,35,39), (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( (1,7,3,5)(2,6,4,8)(9,13,11,15)(10,16,12,14)(17,45,23,51)(18,52,24,46)(19,47,25,41)(20,42,26,48)(21,49,27,43)(22,44,28,50)(29,80,35,86)(30,87,36,81)(31,82,37,88)(32,77,38,83)(33,84,39,78)(34,79,40,85)(53,93,59,99)(54,100,60,94)(55,95,61,89)(56,90,62,96)(57,97,63,91)(58,92,64,98)(65,112,71,106)(66,107,72,101)(67,102,73,108)(68,109,74,103)(69,104,75,110)(70,111,76,105), (1,14,3,16)(2,11,4,9)(5,10,7,12)(6,13,8,15)(17,110,23,104)(18,70,24,76)(19,112,25,106)(20,72,26,66)(21,102,27,108)(22,74,28,68)(29,61,35,55)(30,90,36,96)(31,63,37,57)(32,92,38,98)(33,53,39,59)(34,94,40,100)(41,71,47,65)(42,107,48,101)(43,73,49,67)(44,109,50,103)(45,75,51,69)(46,111,52,105)(54,79,60,85)(56,81,62,87)(58,83,64,77)(78,93,84,99)(80,95,86,89)(82,97,88,91), (1,23,19,63,27,55,59)(2,56,64,24,60,28,20)(3,17,25,57,21,61,53)(4,62,58,18,54,22,26)(5,45,41,97,49,89,93)(6,90,98,46,94,50,42)(7,51,47,91,43,95,99)(8,96,92,52,100,44,48)(9,87,83,70,79,74,66)(10,75,71,88,67,80,84)(11,81,77,76,85,68,72)(12,69,65,82,73,86,78)(13,36,32,111,40,103,107)(14,104,112,37,108,29,33)(15,30,38,105,34,109,101)(16,110,106,31,102,35,39), (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([[(1,7,3,5),(2,6,4,8),(9,13,11,15),(10,16,12,14),(17,45,23,51),(18,52,24,46),(19,47,25,41),(20,42,26,48),(21,49,27,43),(22,44,28,50),(29,80,35,86),(30,87,36,81),(31,82,37,88),(32,77,38,83),(33,84,39,78),(34,79,40,85),(53,93,59,99),(54,100,60,94),(55,95,61,89),(56,90,62,96),(57,97,63,91),(58,92,64,98),(65,112,71,106),(66,107,72,101),(67,102,73,108),(68,109,74,103),(69,104,75,110),(70,111,76,105)], [(1,14,3,16),(2,11,4,9),(5,10,7,12),(6,13,8,15),(17,110,23,104),(18,70,24,76),(19,112,25,106),(20,72,26,66),(21,102,27,108),(22,74,28,68),(29,61,35,55),(30,90,36,96),(31,63,37,57),(32,92,38,98),(33,53,39,59),(34,94,40,100),(41,71,47,65),(42,107,48,101),(43,73,49,67),(44,109,50,103),(45,75,51,69),(46,111,52,105),(54,79,60,85),(56,81,62,87),(58,83,64,77),(78,93,84,99),(80,95,86,89),(82,97,88,91)], [(1,23,19,63,27,55,59),(2,56,64,24,60,28,20),(3,17,25,57,21,61,53),(4,62,58,18,54,22,26),(5,45,41,97,49,89,93),(6,90,98,46,94,50,42),(7,51,47,91,43,95,99),(8,96,92,52,100,44,48),(9,87,83,70,79,74,66),(10,75,71,88,67,80,84),(11,81,77,76,85,68,72),(12,69,65,82,73,86,78),(13,36,32,111,40,103,107),(14,104,112,37,108,29,33),(15,30,38,105,34,109,101),(16,110,106,31,102,35,39)], [(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 Q8.2F7 in GL8(𝔽337)

 336 336 0 0 0 0 0 0 2 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 0 0 0 0 0 0 0 1
,
 151 49 0 0 0 0 0 0 133 186 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
,
 53 140 0 0 0 0 0 0 174 284 0 0 0 0 0 0 0 0 65 0 0 65 209 272 0 0 65 65 209 0 272 0 0 0 274 0 272 65 272 0 0 0 0 65 272 65 0 209 0 0 0 65 0 274 272 272 0 0 65 274 272 0 0 272

`G:=sub<GL(8,GF(337))| [336,2,0,0,0,0,0,0,336,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[151,133,0,0,0,0,0,0,49,186,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[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],[53,174,0,0,0,0,0,0,140,284,0,0,0,0,0,0,0,0,65,65,274,0,0,65,0,0,0,65,0,65,65,274,0,0,0,209,272,272,0,272,0,0,65,0,65,65,274,0,0,0,209,272,272,0,272,0,0,0,272,0,0,209,272,272] >;`

Q8.2F7 in GAP, Magma, Sage, TeX

`Q_8._2F_7`
`% in TeX`

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

`G:=SmallGroup(336,21);`
`// by ID`

`G=gap.SmallGroup(336,21);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,169,151,867,441,69,10373,1745]);`
`// Polycyclic`

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