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## G = C8⋊D7order 112 = 24·7

### 3rd semidirect product of C8 and D7 acting via D7/C7=C2

Aliases: C83D7, C564C2, D14.C4, Dic7.C4, C71M4(2), C4.13D14, C28.13C22, C7⋊C84C2, C2.3(C4×D7), C14.2(C2×C4), (C4×D7).2C2, SmallGroup(112,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C8⋊D7
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C8⋊D7
 Lower central C7 — C14 — C8⋊D7
 Upper central C1 — C4 — C8

Generators and relations for C8⋊D7
G = < a,b,c | a8=b7=c2=1, ab=ba, cac=a5, cbc=b-1 >

Smallest permutation representation of C8⋊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)
(1 49 17 33 44 13 31)(2 50 18 34 45 14 32)(3 51 19 35 46 15 25)(4 52 20 36 47 16 26)(5 53 21 37 48 9 27)(6 54 22 38 41 10 28)(7 55 23 39 42 11 29)(8 56 24 40 43 12 30)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 53)(10 50)(11 55)(12 52)(13 49)(14 54)(15 51)(16 56)(17 44)(18 41)(19 46)(20 43)(21 48)(22 45)(23 42)(24 47)(34 38)(36 40)```

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

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

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

C8⋊D7 is a maximal subgroup of
D28.2C4  D7×M4(2)  D28.C4  D8⋊D7  D56⋊C2  SD16⋊D7  Q16⋊D7  C8⋊F7  C28.32D6  D42.C4  C56⋊S3
C8⋊D7 is a maximal quotient of
Dic7⋊C8  C56⋊C4  D14⋊C8  C28.32D6  D42.C4  C56⋊S3

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 D7 M4(2) D14 C4×D7 C8⋊D7 kernel C8⋊D7 C7⋊C8 C56 C4×D7 Dic7 D14 C8 C7 C4 C2 C1 # reps 1 1 1 1 2 2 3 2 3 6 12

Matrix representation of C8⋊D7 in GL2(𝔽13) generated by

 4 12 8 9
,
 6 6 4 2
,
 2 4 9 11
`G:=sub<GL(2,GF(13))| [4,8,12,9],[6,4,6,2],[2,9,4,11] >;`

C8⋊D7 in GAP, Magma, Sage, TeX

`C_8\rtimes D_7`
`% in TeX`

`G:=Group("C8:D7");`
`// GroupNames label`

`G:=SmallGroup(112,4);`
`// by ID`

`G=gap.SmallGroup(112,4);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-7,101,26,42,2404]);`
`// Polycyclic`

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

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