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## G = D4⋊2D7order 112 = 24·7

### The semidirect product of D4 and D7 acting through Inn(D4)

Aliases: D42D7, D4Dic7, C4.5D14, Dic143C2, C28.5C22, C14.6C23, C22.1D14, D14.2C22, Dic7.4C22, (C4×D7)⋊2C2, (C7×D4)⋊3C2, C72(C4○D4), C7⋊D42C2, (C2×C14).C22, (C2×Dic7)⋊3C2, C2.7(C22×D7), SmallGroup(112,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D4⋊2D7
 Chief series C1 — C7 — C14 — D14 — C4×D7 — D4⋊2D7
 Lower central C7 — C14 — D4⋊2D7
 Upper central C1 — C2 — D4

Generators and relations for D42D7
G = < a,b,c,d | a4=b2=c7=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Character table of D42D7

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A 7B 7C 14A 14B 14C 14D 14E 14F 14G 14H 14I 28A 28B 28C size 1 1 2 2 14 2 7 7 14 14 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 2 2 0 2 0 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ10 2 2 -2 2 0 -2 0 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D14 ρ11 2 2 2 2 0 2 0 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ12 2 2 2 2 0 2 0 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ13 2 2 2 -2 0 -2 0 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 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 -2 -2 0 2 0 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -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 -2 -2 0 2 0 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -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 -2 -2 0 2 0 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ74-ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ17 2 2 2 -2 0 -2 0 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ18 2 2 2 -2 0 -2 0 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D14 ρ19 2 2 -2 2 0 -2 0 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -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 -2 2 0 -2 0 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ21 2 -2 0 0 0 0 -2i 2i 0 0 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 2i -2i 0 0 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 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ24 4 -4 0 0 0 0 0 0 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ25 4 -4 0 0 0 0 0 0 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D42D7
On 56 points
Generators in S56
```(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 48)(2 49)(3 43)(4 44)(5 45)(6 46)(7 47)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(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 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)| (1,41,13,34)(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), (1,48)(2,49)(3,43)(4,44)(5,45)(6,46)(7,47)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(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)(29,31)(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( (1,41,13,34)(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), (1,48)(2,49)(3,43)(4,44)(5,45)(6,46)(7,47)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(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)(29,31)(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([(1,41,13,34),(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)], [(1,48),(2,49),(3,43),(4,44),(5,45),(6,46),(7,47),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(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),(29,31),(32,35),(33,34),(36,38),(39,42),(40,41),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53)])`

Matrix representation of D42D7 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 17 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 0 12 0 0 17 0
,
 0 1 0 0 28 7 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 28
`G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,17,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,0,17,0,0,12,0],[0,28,0,0,1,7,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,28] >;`

D42D7 in GAP, Magma, Sage, TeX

`D_4\rtimes_2D_7`
`% in TeX`

`G:=Group("D4:2D7");`
`// GroupNames label`

`G:=SmallGroup(112,32);`
`// by ID`

`G=gap.SmallGroup(112,32);`
`# by ID`

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

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

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