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## G = Q8.D21order 336 = 24·3·7

### The non-split extension by Q8 of D21 acting via D21/C7=S3

Aliases: Q8.D21, C14.1S4, C7⋊CSU2(𝔽3), SL2(𝔽3).D7, C2.2(C7⋊S4), (C7×Q8).1S3, (C7×SL2(𝔽3)).1C2, SmallGroup(336,118)

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=1, b2=d2=a2, bab-1=a-1, cac-1=b, dad-1=a-1b, cbc-1=ab, dbd-1=a2b, dcd-1=c-1 >

Character table of Q8.D21

 class 1 2 3 4A 4B 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 8 6 84 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 -1 2 0 -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 2 2 0 2 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 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 2 2 0 2 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 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 ρ6 2 2 2 2 0 2 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 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 ρ7 2 2 -1 2 0 -1 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ32ζ74-ζ32ζ73-ζ73 -ζ32ζ75+ζ32ζ72-ζ75 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 -ζ3ζ76+ζ3ζ7-ζ76 ζ3ζ76-ζ3ζ7-ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ32ζ74+ζ32ζ73-ζ74 ζ3ζ76-ζ3ζ7-ζ7 -ζ3ζ76+ζ3ζ7-ζ76 ζ32ζ74-ζ32ζ73-ζ73 -ζ32ζ75+ζ32ζ72-ζ75 -ζ3ζ75+ζ3ζ72-ζ75 orthogonal lifted from D21 ρ8 2 2 -1 2 0 -1 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ3ζ76+ζ3ζ7-ζ76 -ζ32ζ74+ζ32ζ73-ζ74 ζ3ζ76-ζ3ζ7-ζ7 ζ32ζ74-ζ32ζ73-ζ73 -ζ32ζ75+ζ32ζ72-ζ75 -ζ3ζ75+ζ3ζ72-ζ75 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ3ζ76-ζ3ζ7-ζ7 -ζ3ζ75+ζ3ζ72-ζ75 -ζ32ζ75+ζ32ζ72-ζ75 -ζ3ζ76+ζ3ζ7-ζ76 -ζ32ζ74+ζ32ζ73-ζ74 ζ32ζ74-ζ32ζ73-ζ73 orthogonal lifted from D21 ρ9 2 2 -1 2 0 -1 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ3ζ75+ζ3ζ72-ζ75 -ζ3ζ76+ζ3ζ7-ζ76 -ζ32ζ75+ζ32ζ72-ζ75 ζ3ζ76-ζ3ζ7-ζ7 ζ32ζ74-ζ32ζ73-ζ73 -ζ32ζ74+ζ32ζ73-ζ74 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ32ζ75+ζ32ζ72-ζ75 -ζ32ζ74+ζ32ζ73-ζ74 ζ32ζ74-ζ32ζ73-ζ73 -ζ3ζ75+ζ3ζ72-ζ75 -ζ3ζ76+ζ3ζ7-ζ76 ζ3ζ76-ζ3ζ7-ζ7 orthogonal lifted from D21 ρ10 2 2 -1 2 0 -1 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 ζ32ζ74-ζ32ζ73-ζ73 -ζ32ζ75+ζ32ζ72-ζ75 ζ3ζ76-ζ3ζ7-ζ7 -ζ3ζ76+ζ3ζ7-ζ76 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ32ζ74-ζ32ζ73-ζ73 -ζ3ζ76+ζ3ζ7-ζ76 ζ3ζ76-ζ3ζ7-ζ7 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 -ζ32ζ75+ζ32ζ72-ζ75 orthogonal lifted from D21 ρ11 2 2 -1 2 0 -1 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ3ζ76-ζ3ζ7-ζ7 ζ32ζ74-ζ32ζ73-ζ73 -ζ3ζ76+ζ3ζ7-ζ76 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 -ζ32ζ75+ζ32ζ72-ζ75 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ3ζ76+ζ3ζ7-ζ76 -ζ32ζ75+ζ32ζ72-ζ75 -ζ3ζ75+ζ3ζ72-ζ75 ζ3ζ76-ζ3ζ7-ζ7 ζ32ζ74-ζ32ζ73-ζ73 -ζ32ζ74+ζ32ζ73-ζ74 orthogonal lifted from D21 ρ12 2 2 -1 2 0 -1 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ32ζ75+ζ32ζ72-ζ75 ζ3ζ76-ζ3ζ7-ζ7 -ζ3ζ75+ζ3ζ72-ζ75 -ζ3ζ76+ζ3ζ7-ζ76 -ζ32ζ74+ζ32ζ73-ζ74 ζ32ζ74-ζ32ζ73-ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ3ζ75+ζ3ζ72-ζ75 ζ32ζ74-ζ32ζ73-ζ73 -ζ32ζ74+ζ32ζ73-ζ74 -ζ32ζ75+ζ32ζ72-ζ75 ζ3ζ76-ζ3ζ7-ζ7 -ζ3ζ76+ζ3ζ7-ζ76 orthogonal lifted from D21 ρ13 2 -2 -1 0 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 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ14 2 -2 -1 0 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 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ15 3 3 0 -1 -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 0 -1 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 1 0 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 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ18 4 -4 -2 0 0 2 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 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 symplectic faithful, Schur index 2 ρ19 4 -4 -2 0 0 2 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 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 symplectic faithful, Schur index 2 ρ20 4 -4 -2 0 0 2 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 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 symplectic faithful, Schur index 2 ρ21 4 -4 1 0 0 -1 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 0 0 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 -ζ3ζ75+ζ3ζ72+ζ72 -ζ3ζ76+ζ3ζ7+ζ7 -ζ32ζ75+ζ32ζ72+ζ72 ζ3ζ76-ζ3ζ7+ζ76 ζ32ζ74-ζ32ζ73+ζ74 -ζ32ζ74+ζ32ζ73+ζ73 0 0 0 -ζ3ζ75+ζ3ζ72-ζ75 ζ32ζ74-ζ32ζ73-ζ73 -ζ32ζ74+ζ32ζ73-ζ74 -ζ32ζ75+ζ32ζ72-ζ75 ζ3ζ76-ζ3ζ7-ζ7 -ζ3ζ76+ζ3ζ7-ζ76 symplectic faithful, Schur index 2 ρ22 4 -4 1 0 0 -1 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 0 0 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 ζ32ζ74-ζ32ζ73+ζ74 -ζ32ζ75+ζ32ζ72+ζ72 -ζ32ζ74+ζ32ζ73+ζ73 -ζ3ζ75+ζ3ζ72+ζ72 -ζ3ζ76+ζ3ζ7+ζ7 ζ3ζ76-ζ3ζ7+ζ76 0 0 0 ζ32ζ74-ζ32ζ73-ζ73 -ζ3ζ76+ζ3ζ7-ζ76 ζ3ζ76-ζ3ζ7-ζ7 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 -ζ32ζ75+ζ32ζ72-ζ75 symplectic faithful, Schur index 2 ρ23 4 -4 1 0 0 -1 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 0 0 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 ζ3ζ76-ζ3ζ7+ζ76 ζ32ζ74-ζ32ζ73+ζ74 -ζ3ζ76+ζ3ζ7+ζ7 -ζ32ζ74+ζ32ζ73+ζ73 -ζ3ζ75+ζ3ζ72+ζ72 -ζ32ζ75+ζ32ζ72+ζ72 0 0 0 ζ3ζ76-ζ3ζ7-ζ7 -ζ3ζ75+ζ3ζ72-ζ75 -ζ32ζ75+ζ32ζ72-ζ75 -ζ3ζ76+ζ3ζ7-ζ76 -ζ32ζ74+ζ32ζ73-ζ74 ζ32ζ74-ζ32ζ73-ζ73 symplectic faithful, Schur index 2 ρ24 4 -4 1 0 0 -1 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 0 0 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 -ζ32ζ75+ζ32ζ72+ζ72 ζ3ζ76-ζ3ζ7+ζ76 -ζ3ζ75+ζ3ζ72+ζ72 -ζ3ζ76+ζ3ζ7+ζ7 -ζ32ζ74+ζ32ζ73+ζ73 ζ32ζ74-ζ32ζ73+ζ74 0 0 0 -ζ32ζ75+ζ32ζ72-ζ75 -ζ32ζ74+ζ32ζ73-ζ74 ζ32ζ74-ζ32ζ73-ζ73 -ζ3ζ75+ζ3ζ72-ζ75 -ζ3ζ76+ζ3ζ7-ζ76 ζ3ζ76-ζ3ζ7-ζ7 symplectic faithful, Schur index 2 ρ25 4 -4 1 0 0 -1 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 0 0 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 -ζ32ζ74+ζ32ζ73+ζ73 -ζ3ζ75+ζ3ζ72+ζ72 ζ32ζ74-ζ32ζ73+ζ74 -ζ32ζ75+ζ32ζ72+ζ72 ζ3ζ76-ζ3ζ7+ζ76 -ζ3ζ76+ζ3ζ7+ζ7 0 0 0 -ζ32ζ74+ζ32ζ73-ζ74 ζ3ζ76-ζ3ζ7-ζ7 -ζ3ζ76+ζ3ζ7-ζ76 ζ32ζ74-ζ32ζ73-ζ73 -ζ32ζ75+ζ32ζ72-ζ75 -ζ3ζ75+ζ3ζ72-ζ75 symplectic faithful, Schur index 2 ρ26 4 -4 1 0 0 -1 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 0 0 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 -ζ3ζ76+ζ3ζ7+ζ7 -ζ32ζ74+ζ32ζ73+ζ73 ζ3ζ76-ζ3ζ7+ζ76 ζ32ζ74-ζ32ζ73+ζ74 -ζ32ζ75+ζ32ζ72+ζ72 -ζ3ζ75+ζ3ζ72+ζ72 0 0 0 -ζ3ζ76+ζ3ζ7-ζ76 -ζ32ζ75+ζ32ζ72-ζ75 -ζ3ζ75+ζ3ζ72-ζ75 ζ3ζ76-ζ3ζ7-ζ7 ζ32ζ74-ζ32ζ73-ζ73 -ζ32ζ74+ζ32ζ73-ζ74 symplectic faithful, Schur index 2 ρ27 6 6 0 -2 0 0 3ζ75+3ζ72 3ζ76+3ζ7 3ζ74+3ζ73 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 ρ28 6 6 0 -2 0 0 3ζ76+3ζ7 3ζ74+3ζ73 3ζ75+3ζ72 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 -2 0 0 3ζ74+3ζ73 3ζ75+3ζ72 3ζ76+3ζ7 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

Smallest permutation representation of Q8.D21
On 112 points
Generators in S112
```(1 83 21 57)(2 77 15 51)(3 71 16 66)(4 86 17 60)(5 80 18 54)(6 74 19 69)(7 89 20 63)(8 31 25 94)(9 46 26 109)(10 40 27 103)(11 34 28 97)(12 49 22 112)(13 43 23 106)(14 37 24 100)(29 36 92 99)(30 107 93 44)(32 39 95 102)(33 110 96 47)(35 42 98 105)(38 45 101 108)(41 48 104 111)(50 90 76 64)(52 59 78 85)(53 72 79 67)(55 62 81 88)(56 75 82 70)(58 65 84 91)(61 68 87 73)
(1 76 21 50)(2 91 15 65)(3 85 16 59)(4 79 17 53)(5 73 18 68)(6 88 19 62)(7 82 20 56)(8 45 25 108)(9 39 26 102)(10 33 27 96)(11 48 28 111)(12 42 22 105)(13 36 23 99)(14 30 24 93)(29 106 92 43)(31 38 94 101)(32 109 95 46)(34 41 97 104)(35 112 98 49)(37 44 100 107)(40 47 103 110)(51 58 77 84)(52 71 78 66)(54 61 80 87)(55 74 81 69)(57 64 83 90)(60 67 86 72)(63 70 89 75)
(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)
(1 27 21 10)(2 26 15 9)(3 25 16 8)(4 24 17 14)(5 23 18 13)(6 22 19 12)(7 28 20 11)(29 54 92 80)(30 53 93 79)(31 52 94 78)(32 51 95 77)(33 50 96 76)(34 70 97 75)(35 69 98 74)(36 68 99 73)(37 67 100 72)(38 66 101 71)(39 65 102 91)(40 64 103 90)(41 63 104 89)(42 62 105 88)(43 61 106 87)(44 60 107 86)(45 59 108 85)(46 58 109 84)(47 57 110 83)(48 56 111 82)(49 55 112 81)```

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

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

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

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

 1 0 0 0 0 1 0 0 0 0 312 35 0 0 11 25
,
 1 0 0 0 0 1 0 0 0 0 36 326 0 0 26 301
,
 63 234 0 0 103 297 0 0 0 0 336 1 0 0 336 0
,
 123 8 0 0 131 214 0 0 0 0 142 119 0 0 261 195
`G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,312,11,0,0,35,25],[1,0,0,0,0,1,0,0,0,0,36,26,0,0,326,301],[63,103,0,0,234,297,0,0,0,0,336,336,0,0,1,0],[123,131,0,0,8,214,0,0,0,0,142,261,0,0,119,195] >;`

Q8.D21 in GAP, Magma, Sage, TeX

`Q_8.D_{21}`
`% in TeX`

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

`G:=SmallGroup(336,118);`
`// by ID`

`G=gap.SmallGroup(336,118);`
`# by ID`

`G:=PCGroup([6,-2,-3,-7,-2,2,-2,1008,49,650,2019,3033,117,1264,1900,202,88]);`
`// Polycyclic`

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

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