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## G = D8⋊D7order 224 = 25·7

### 2nd semidirect product of D8 and D7 acting via D7/C7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D8⋊D7
 Chief series C1 — C7 — C14 — C28 — C4×D7 — D4×D7 — D8⋊D7
 Lower central C7 — C14 — C28 — D8⋊D7
 Upper central C1 — C2 — C4 — D8

Generators and relations for D8⋊D7
G = < a,b,c,d | a8=b2=c7=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 350 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×4], C4, C4 [×2], C22 [×6], C7, C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, D7 [×2], C14, C14 [×2], M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, Dic7, Dic7, C28, D14, D14 [×3], C2×C14 [×2], C8⋊C22, C7⋊C8, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4 [×2], C7×D4 [×2], C22×D7, C8⋊D7, C56⋊C2, D4⋊D7, D4.D7, C7×D8, D4×D7, D42D7, D8⋊D7
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], C8⋊C22, C22×D7, D4×D7, D8⋊D7

Smallest permutation representation of D8⋊D7
On 56 points
Generators in S56
```(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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(18 24)(19 23)(20 22)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(42 48)(43 47)(44 46)(49 55)(50 54)(51 53)
(1 26 37 10 21 52 41)(2 27 38 11 22 53 42)(3 28 39 12 23 54 43)(4 29 40 13 24 55 44)(5 30 33 14 17 56 45)(6 31 34 15 18 49 46)(7 32 35 16 19 50 47)(8 25 36 9 20 51 48)
(1 41)(2 46)(3 43)(4 48)(5 45)(6 42)(7 47)(8 44)(9 13)(11 15)(17 33)(18 38)(19 35)(20 40)(21 37)(22 34)(23 39)(24 36)(25 55)(26 52)(27 49)(28 54)(29 51)(30 56)(31 53)(32 50)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 7A 7B 7C 8A 8B 14A 14B 14C 14D ··· 14I 28A 28B 28C 56A ··· 56F order 1 2 2 2 2 2 4 4 4 7 7 7 8 8 14 14 14 14 ··· 14 28 28 28 56 ··· 56 size 1 1 4 4 14 28 2 14 28 2 2 2 4 28 2 2 2 8 ··· 8 4 4 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 C8⋊C22 D4×D7 D8⋊D7 kernel D8⋊D7 C8⋊D7 C56⋊C2 D4⋊D7 D4.D7 C7×D8 D4×D7 D4⋊2D7 Dic7 D14 D8 C8 D4 C7 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 3 3 6 1 3 6

Matrix representation of D8⋊D7 in GL4(𝔽113) generated by

 0 0 43 49 0 0 71 5 17 82 103 95 75 101 18 10
,
 1 0 0 0 0 1 0 0 51 39 112 0 74 0 0 112
,
 34 1 0 0 111 103 0 0 0 0 0 1 0 0 112 24
,
 103 112 0 0 99 10 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(113))| [0,0,17,75,0,0,82,101,43,71,103,18,49,5,95,10],[1,0,51,74,0,1,39,0,0,0,112,0,0,0,0,112],[34,111,0,0,1,103,0,0,0,0,0,112,0,0,1,24],[103,99,0,0,112,10,0,0,0,0,0,1,0,0,1,0] >;`

D8⋊D7 in GAP, Magma, Sage, TeX

`D_8\rtimes D_7`
`% in TeX`

`G:=Group("D8:D7");`
`// GroupNames label`

`G:=SmallGroup(224,106);`
`// by ID`

`G=gap.SmallGroup(224,106);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,362,116,297,159,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^7=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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