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## G = Q8⋊2D7order 112 = 24·7

### The semidirect product of Q8 and D7 acting through Inn(Q8)

Aliases: Q82D7, Q8Dic7, D284C2, C4.7D14, C28.7C22, C14.8C23, D14.3C22, Dic7.5C22, (C4×D7)⋊3C2, C73(C4○D4), (C7×Q8)⋊3C2, C2.9(C22×D7), SmallGroup(112,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Q8⋊2D7
 Chief series C1 — C7 — C14 — D14 — C4×D7 — Q8⋊2D7
 Lower central C7 — C14 — Q8⋊2D7
 Upper central C1 — C2 — Q8

Generators and relations for Q82D7
G = < a,b,c,d | a4=c7=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Character table of Q82D7

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A 7B 7C 14A 14B 14C 28A 28B 28C 28D 28E 28F 28G 28H 28I size 1 1 14 14 14 2 2 2 7 7 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 0 0 0 -2 -2 2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ10 2 2 0 0 0 2 -2 -2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ11 2 2 0 0 0 2 2 2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ12 2 2 0 0 0 -2 2 -2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D14 ρ13 2 2 0 0 0 -2 -2 2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D14 ρ14 2 2 0 0 0 -2 2 -2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D14 ρ15 2 2 0 0 0 2 -2 -2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D14 ρ16 2 2 0 0 0 2 2 2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ17 2 2 0 0 0 2 2 2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ18 2 2 0 0 0 -2 2 -2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ19 2 2 0 0 0 -2 -2 2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D14 ρ20 2 2 0 0 0 2 -2 -2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D14 ρ21 2 -2 0 0 0 0 0 0 -2i 2i 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 0 0 0 0 0 0 2i -2i 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 0 0 0 0 0 0 0 0 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2 ρ24 4 -4 0 0 0 0 0 0 0 0 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2 ρ25 4 -4 0 0 0 0 0 0 0 0 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of Q82D7
On 56 points
Generators in S56
```(1 27 13 20)(2 28 14 21)(3 22 8 15)(4 23 9 16)(5 24 10 17)(6 25 11 18)(7 26 12 19)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)
(1 41 13 34)(2 42 14 35)(3 36 8 29)(4 37 9 30)(5 38 10 31)(6 39 11 32)(7 40 12 33)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)
(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)```

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

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

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

Q82D7 is a maximal subgroup of
D56⋊C2  SD163D7  Q16⋊D7  Q8.D14  Q8.10D14  D7×C4○D4  D48D14  Q83F7  Dic7.2A4  Q8.F7  D28⋊S3  D14.D6  Q83D21
Q82D7 is a maximal quotient of
C28.3Q8  C4⋊C47D7  D28⋊C4  D14.5D4  C4⋊D28  C4⋊C4⋊D7  Q8×Dic7  D143Q8  C28.23D4  D28⋊S3  D14.D6  Q83D21

Matrix representation of Q82D7 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 1 13 0 0 11 28
,
 28 0 0 0 0 28 0 0 0 0 17 0 0 0 13 12
,
 0 1 0 0 28 18 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 11 28
`G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,1,11,0,0,13,28],[28,0,0,0,0,28,0,0,0,0,17,13,0,0,0,12],[0,28,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,11,0,0,0,28] >;`

Q82D7 in GAP, Magma, Sage, TeX

`Q_8\rtimes_2D_7`
`% in TeX`

`G:=Group("Q8:2D7");`
`// GroupNames label`

`G:=SmallGroup(112,34);`
`// by ID`

`G=gap.SmallGroup(112,34);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-7,46,182,97,42,2404]);`
`// Polycyclic`

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

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