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## G = D4⋊D7order 112 = 24·7

### The semidirect product of D4 and D7 acting via D7/C7=C2

Aliases: D4⋊D7, C72D8, D282C2, C4.1D14, C14.7D4, C28.1C22, C7⋊C81C2, (C7×D4)⋊1C2, C2.4(C7⋊D4), SmallGroup(112,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D4⋊D7
 Chief series C1 — C7 — C14 — C28 — D28 — D4⋊D7
 Lower central C7 — C14 — C28 — D4⋊D7
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4⋊D7
G = < a,b,c,d | a4=b2=c7=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >

Character table of D4⋊D7

 class 1 2A 2B 2C 4 7A 7B 7C 8A 8B 14A 14B 14C 14D 14E 14F 14G 14H 14I 28A 28B 28C size 1 1 4 28 2 2 2 2 14 14 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 trivial ρ2 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 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 linear of order 2 ρ5 2 2 0 0 -2 2 2 2 0 0 2 2 2 0 0 0 0 0 0 -2 -2 -2 orthogonal lifted from D4 ρ6 2 -2 0 0 0 2 2 2 √2 -√2 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ7 2 2 2 0 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ8 2 2 2 0 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ9 2 2 -2 0 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ74-ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D14 ρ10 2 2 -2 0 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ76-ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ11 2 2 2 0 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ12 2 2 -2 0 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ75-ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D14 ρ13 2 -2 0 0 0 2 2 2 -√2 √2 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ14 2 2 0 0 -2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76-ζ7 -ζ75+ζ72 ζ75-ζ72 -ζ76+ζ7 ζ74-ζ73 -ζ74+ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 complex lifted from C7⋊D4 ρ15 2 2 0 0 -2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74-ζ73 ζ76-ζ7 -ζ76+ζ7 -ζ74+ζ73 ζ75-ζ72 -ζ75+ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 complex lifted from C7⋊D4 ρ16 2 2 0 0 -2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76+ζ7 ζ75-ζ72 -ζ75+ζ72 ζ76-ζ7 -ζ74+ζ73 ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 complex lifted from C7⋊D4 ρ17 2 2 0 0 -2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74+ζ73 -ζ76+ζ7 ζ76-ζ7 ζ74-ζ73 -ζ75+ζ72 ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 complex lifted from C7⋊D4 ρ18 2 2 0 0 -2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75-ζ72 ζ74-ζ73 -ζ74+ζ73 -ζ75+ζ72 -ζ76+ζ7 ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 complex lifted from C7⋊D4 ρ19 2 2 0 0 -2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75+ζ72 -ζ74+ζ73 ζ74-ζ73 ζ75-ζ72 ζ76-ζ7 -ζ76+ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 complex lifted from C7⋊D4 ρ20 4 -4 0 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ21 4 -4 0 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ22 4 -4 0 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of D4⋊D7
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 55)(2 56)(3 50)(4 51)(5 52)(6 53)(7 54)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)
(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 45)(30 44)(31 43)(32 49)(33 48)(34 47)(35 46)(36 52)(37 51)(38 50)(39 56)(40 55)(41 54)(42 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), (1,55)(2,56)(3,50)(4,51)(5,52)(6,53)(7,54)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42), (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)(29,45)(30,44)(31,43)(32,49)(33,48)(34,47)(35,46)(36,52)(37,51)(38,50)(39,56)(40,55)(41,54)(42,53)>;`

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

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

D4⋊D7 is a maximal subgroup of
D7×D8  D8⋊D7  D56⋊C2  SD163D7  D4.D14  D4⋊D14  D4.8D14  D4⋊F7  C21⋊D8  C7⋊D24  D4⋊D21
D4⋊D7 is a maximal quotient of
C28.Q8  C14.D8  C7⋊D16  D8.D7  C7⋊SD32  C7⋊Q32  D4⋊Dic7  C21⋊D8  C7⋊D24  D4⋊D21

Matrix representation of D4⋊D7 in GL4(𝔽113) generated by

 0 1 0 0 112 0 0 0 0 0 1 0 0 0 0 1
,
 82 31 0 0 31 31 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 112 1 0 0 87 25
,
 1 0 0 0 0 112 0 0 0 0 112 0 0 0 87 1
`G:=sub<GL(4,GF(113))| [0,112,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[82,31,0,0,31,31,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,87,0,0,1,25],[1,0,0,0,0,112,0,0,0,0,112,87,0,0,0,1] >;`

D4⋊D7 in GAP, Magma, Sage, TeX

`D_4\rtimes D_7`
`% in TeX`

`G:=Group("D4:D7");`
`// GroupNames label`

`G:=SmallGroup(112,14);`
`// by ID`

`G=gap.SmallGroup(112,14);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-7,61,182,97,42,2404]);`
`// Polycyclic`

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

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