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## G = Q8⋊D21order 336 = 24·3·7

### The semidirect product of Q8 and D21 acting via D21/C7=S3

Aliases: Q8⋊D21, C14.2S4, C7⋊GL2(𝔽3), SL2(𝔽3)⋊D7, C2.3(C7⋊S4), (C7×Q8)⋊1S3, (C7×SL2(𝔽3))⋊1C2, SmallGroup(336,119)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C7×SL2(𝔽3) — Q8⋊D21
 Chief series C1 — C2 — Q8 — C7×Q8 — C7×SL2(𝔽3) — Q8⋊D21
 Lower central C7×SL2(𝔽3) — Q8⋊D21
 Upper central C1 — C2

Generators and relations for Q8⋊D21
G = < a,b,c,d | a4=c21=d2=1, b2=a2, bab-1=a-1, cac-1=b, dad=a-1b, cbc-1=ab, dbd=a2b, dcd=c-1 >

84C2
4C3
3C4
42C22
4C6
28S3
28S3
12D7
4C21
21D4
21C8
28D6
3C28
6D14
4D21
4D21
4C42
21SD16
3D28
4D42

Character table of Q8⋊D21

 class 1 2A 2B 3 4 6 7A 7B 7C 8A 8B 14A 14B 14C 21A 21B 21C 21D 21E 21F 28A 28B 28C 42A 42B 42C 42D 42E 42F size 1 1 84 8 6 8 2 2 2 42 42 2 2 2 8 8 8 8 8 8 12 12 12 8 8 8 8 8 8 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 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 1 1 1 linear of order 2 ρ3 2 2 0 -1 2 -1 2 2 2 0 0 2 2 2 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 0 2 2 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 orthogonal lifted from D7 ρ5 2 2 0 2 2 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 orthogonal lifted from D7 ρ6 2 2 0 2 2 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 orthogonal lifted from D7 ρ7 2 2 0 -1 2 -1 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ3ζ75-ζ3ζ72-ζ72 -ζ32ζ76+ζ32ζ7-ζ76 -ζ3ζ75+ζ3ζ72-ζ75 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ74+ζ3ζ73-ζ74 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ3ζ75+ζ3ζ72-ζ75 -ζ3ζ74+ζ3ζ73-ζ74 -ζ32ζ74+ζ32ζ73-ζ74 ζ3ζ75-ζ3ζ72-ζ72 -ζ32ζ76+ζ32ζ7-ζ76 ζ32ζ76-ζ32ζ7-ζ7 orthogonal lifted from D21 ρ8 2 2 0 -1 2 -1 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ3ζ74+ζ3ζ73-ζ74 ζ3ζ75-ζ3ζ72-ζ72 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ76+ζ32ζ7-ζ76 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ32ζ74+ζ32ζ73-ζ74 -ζ32ζ76+ζ32ζ7-ζ76 ζ32ζ76-ζ32ζ7-ζ7 -ζ3ζ74+ζ3ζ73-ζ74 ζ3ζ75-ζ3ζ72-ζ72 -ζ3ζ75+ζ3ζ72-ζ75 orthogonal lifted from D21 ρ9 2 2 0 -1 2 -1 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ74+ζ32ζ73-ζ74 -ζ32ζ76+ζ32ζ7-ζ76 -ζ3ζ74+ζ3ζ73-ζ74 ζ3ζ75-ζ3ζ72-ζ72 -ζ3ζ75+ζ3ζ72-ζ75 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ32ζ76+ζ32ζ7-ζ76 -ζ3ζ75+ζ3ζ72-ζ75 ζ3ζ75-ζ3ζ72-ζ72 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ74+ζ3ζ73-ζ74 orthogonal lifted from D21 ρ10 2 2 0 -1 2 -1 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ32ζ76+ζ32ζ7-ζ76 -ζ3ζ74+ζ3ζ73-ζ74 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 ζ3ζ75-ζ3ζ72-ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ32ζ76-ζ32ζ7-ζ7 ζ3ζ75-ζ3ζ72-ζ72 -ζ3ζ75+ζ3ζ72-ζ75 -ζ32ζ76+ζ32ζ7-ζ76 -ζ3ζ74+ζ3ζ73-ζ74 -ζ32ζ74+ζ32ζ73-ζ74 orthogonal lifted from D21 ρ11 2 2 0 -1 2 -1 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 -ζ3ζ74+ζ3ζ73-ζ74 ζ3ζ75-ζ3ζ72-ζ72 -ζ32ζ76+ζ32ζ7-ζ76 ζ32ζ76-ζ32ζ7-ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ3ζ74+ζ3ζ73-ζ74 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ76+ζ32ζ7-ζ76 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 ζ3ζ75-ζ3ζ72-ζ72 orthogonal lifted from D21 ρ12 2 2 0 -1 2 -1 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ3ζ75+ζ3ζ72-ζ75 ζ32ζ76-ζ32ζ7-ζ7 ζ3ζ75-ζ3ζ72-ζ72 -ζ32ζ76+ζ32ζ7-ζ76 -ζ3ζ74+ζ3ζ73-ζ74 -ζ32ζ74+ζ32ζ73-ζ74 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ3ζ75-ζ3ζ72-ζ72 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ74+ζ3ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ76+ζ32ζ7-ζ76 orthogonal lifted from D21 ρ13 2 -2 0 -1 0 1 2 2 2 √-2 -√-2 -2 -2 -2 -1 -1 -1 -1 -1 -1 0 0 0 1 1 1 1 1 1 complex lifted from GL2(𝔽3) ρ14 2 -2 0 -1 0 1 2 2 2 -√-2 √-2 -2 -2 -2 -1 -1 -1 -1 -1 -1 0 0 0 1 1 1 1 1 1 complex lifted from GL2(𝔽3) ρ15 3 3 -1 0 -1 0 3 3 3 1 1 3 3 3 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ16 3 3 1 0 -1 0 3 3 3 -1 -1 3 3 3 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ17 4 -4 0 1 0 -1 4 4 4 0 0 -4 -4 -4 1 1 1 1 1 1 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from GL2(𝔽3) ρ18 4 -4 0 -2 0 2 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 -ζ76-ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ75-ζ72 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 orthogonal faithful, Schur index 2 ρ19 4 -4 0 -2 0 2 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 -ζ74-ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ76-ζ7 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 orthogonal faithful, Schur index 2 ρ20 4 -4 0 -2 0 2 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 -ζ75-ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ74-ζ73 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 orthogonal faithful, Schur index 2 ρ21 4 -4 0 1 0 -1 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 ζ3ζ75-ζ3ζ72+ζ75 ζ3ζ76-ζ3ζ7+ζ76 ζ32ζ75-ζ32ζ72+ζ75 -ζ3ζ76+ζ3ζ7+ζ7 -ζ32ζ74+ζ32ζ73+ζ73 ζ32ζ74-ζ32ζ73+ζ74 0 0 0 ζ3ζ75-ζ3ζ72-ζ72 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ74+ζ3ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ76+ζ32ζ7-ζ76 orthogonal faithful ρ22 4 -4 0 1 0 -1 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 ζ3ζ76-ζ3ζ7+ζ76 ζ32ζ74-ζ32ζ73+ζ74 -ζ3ζ76+ζ3ζ7+ζ7 -ζ32ζ74+ζ32ζ73+ζ73 ζ32ζ75-ζ32ζ72+ζ75 ζ3ζ75-ζ3ζ72+ζ75 0 0 0 -ζ32ζ76+ζ32ζ7-ζ76 -ζ3ζ75+ζ3ζ72-ζ75 ζ3ζ75-ζ3ζ72-ζ72 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ74+ζ3ζ73-ζ74 orthogonal faithful ρ23 4 -4 0 1 0 -1 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 ζ32ζ74-ζ32ζ73+ζ74 ζ3ζ75-ζ3ζ72+ζ75 -ζ32ζ74+ζ32ζ73+ζ73 ζ32ζ75-ζ32ζ72+ζ75 -ζ3ζ76+ζ3ζ7+ζ7 ζ3ζ76-ζ3ζ7+ζ76 0 0 0 -ζ3ζ74+ζ3ζ73-ζ74 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ76+ζ32ζ7-ζ76 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 ζ3ζ75-ζ3ζ72-ζ72 orthogonal faithful ρ24 4 -4 0 1 0 -1 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 ζ32ζ75-ζ32ζ72+ζ75 -ζ3ζ76+ζ3ζ7+ζ7 ζ3ζ75-ζ3ζ72+ζ75 ζ3ζ76-ζ3ζ7+ζ76 ζ32ζ74-ζ32ζ73+ζ74 -ζ32ζ74+ζ32ζ73+ζ73 0 0 0 -ζ3ζ75+ζ3ζ72-ζ75 -ζ3ζ74+ζ3ζ73-ζ74 -ζ32ζ74+ζ32ζ73-ζ74 ζ3ζ75-ζ3ζ72-ζ72 -ζ32ζ76+ζ32ζ7-ζ76 ζ32ζ76-ζ32ζ7-ζ7 orthogonal faithful ρ25 4 -4 0 1 0 -1 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 -ζ32ζ74+ζ32ζ73+ζ73 ζ32ζ75-ζ32ζ72+ζ75 ζ32ζ74-ζ32ζ73+ζ74 ζ3ζ75-ζ3ζ72+ζ75 ζ3ζ76-ζ3ζ7+ζ76 -ζ3ζ76+ζ3ζ7+ζ7 0 0 0 -ζ32ζ74+ζ32ζ73-ζ74 -ζ32ζ76+ζ32ζ7-ζ76 ζ32ζ76-ζ32ζ7-ζ7 -ζ3ζ74+ζ3ζ73-ζ74 ζ3ζ75-ζ3ζ72-ζ72 -ζ3ζ75+ζ3ζ72-ζ75 orthogonal faithful ρ26 4 -4 0 1 0 -1 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 -ζ3ζ76+ζ3ζ7+ζ7 -ζ32ζ74+ζ32ζ73+ζ73 ζ3ζ76-ζ3ζ7+ζ76 ζ32ζ74-ζ32ζ73+ζ74 ζ3ζ75-ζ3ζ72+ζ75 ζ32ζ75-ζ32ζ72+ζ75 0 0 0 ζ32ζ76-ζ32ζ7-ζ7 ζ3ζ75-ζ3ζ72-ζ72 -ζ3ζ75+ζ3ζ72-ζ75 -ζ32ζ76+ζ32ζ7-ζ76 -ζ3ζ74+ζ3ζ73-ζ74 -ζ32ζ74+ζ32ζ73-ζ74 orthogonal faithful ρ27 6 6 0 0 -2 0 3ζ76+3ζ7 3ζ75+3ζ72 3ζ74+3ζ73 0 0 3ζ75+3ζ72 3ζ74+3ζ73 3ζ76+3ζ7 0 0 0 0 0 0 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 0 0 0 0 0 0 orthogonal lifted from C7⋊S4 ρ28 6 6 0 0 -2 0 3ζ75+3ζ72 3ζ74+3ζ73 3ζ76+3ζ7 0 0 3ζ74+3ζ73 3ζ76+3ζ7 3ζ75+3ζ72 0 0 0 0 0 0 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 0 0 0 0 0 0 orthogonal lifted from C7⋊S4 ρ29 6 6 0 0 -2 0 3ζ74+3ζ73 3ζ76+3ζ7 3ζ75+3ζ72 0 0 3ζ76+3ζ7 3ζ75+3ζ72 3ζ74+3ζ73 0 0 0 0 0 0 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 0 0 0 0 0 0 orthogonal lifted from C7⋊S4

Smallest permutation representation of Q8⋊D21
On 56 points
Generators in S56
```(1 54 8 21)(2 48 9 15)(3 42 10 30)(4 36 11 24)(5 51 12 18)(6 45 13 33)(7 39 14 27)(16 23 49 56)(17 43 50 31)(19 26 52 38)(20 46 53 34)(22 29 55 41)(25 32 37 44)(28 35 40 47)
(1 47 8 35)(2 41 9 29)(3 56 10 23)(4 50 11 17)(5 44 12 32)(6 38 13 26)(7 53 14 20)(15 22 48 55)(16 42 49 30)(18 25 51 37)(19 45 52 33)(21 28 54 40)(24 31 36 43)(27 34 39 46)
(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)
(1 5)(2 4)(6 7)(8 12)(9 11)(13 14)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)```

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

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

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

Matrix representation of Q8⋊D21 in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 3 281 0 0 277 334
,
 1 0 0 0 0 1 0 0 0 0 278 335 0 0 56 59
,
 274 20 0 0 257 234 0 0 0 0 56 59 0 0 3 280
,
 247 287 0 0 317 90 0 0 0 0 186 282 0 0 292 151
`G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,3,277,0,0,281,334],[1,0,0,0,0,1,0,0,0,0,278,56,0,0,335,59],[274,257,0,0,20,234,0,0,0,0,56,3,0,0,59,280],[247,317,0,0,287,90,0,0,0,0,186,292,0,0,282,151] >;`

Q8⋊D21 in GAP, Magma, Sage, TeX

`Q_8\rtimes D_{21}`
`% in TeX`

`G:=Group("Q8:D21");`
`// GroupNames label`

`G:=SmallGroup(336,119);`
`// by ID`

`G=gap.SmallGroup(336,119);`
`# by ID`

`G:=PCGroup([6,-2,-3,-7,-2,2,-2,49,650,2019,3033,117,1264,1900,202,88]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^21=d^2=1,b^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,d*a*d=a^-1*b,c*b*c^-1=a*b,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

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