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## G = C8⋊D14order 224 = 25·7

### 1st semidirect product of C8 and D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C8⋊D14
 Chief series C1 — C7 — C14 — C28 — D28 — C2×D28 — C8⋊D14
 Lower central C7 — C14 — C28 — C8⋊D14
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C8⋊D14
G = < a,b,c | a8=b14=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 398 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, M4(2), D8, SD16, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C8⋊C22, C56, Dic14, C4×D7, D28, D28, D28, C7⋊D4, C2×C28, C22×D7, C56⋊C2, D56, C7×M4(2), C2×D28, C4○D28, C8⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8⋊C22, D28, C22×D7, C2×D28, C8⋊D14

Smallest permutation representation of C8⋊D14
On 56 points
Generators in S56
```(1 35 15 47 12 42 25 54)(2 29 16 55 13 36 26 48)(3 37 17 49 14 30 27 56)(4 31 18 43 8 38 28 50)(5 39 19 51 9 32 22 44)(6 33 20 45 10 40 23 52)(7 41 21 53 11 34 24 46)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 21)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 28)(29 39)(30 38)(31 37)(32 36)(33 35)(40 42)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)```

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

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

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

41 conjugacy classes

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

41 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 D28 D28 C8⋊C22 C8⋊D14 kernel C8⋊D14 C56⋊C2 D56 C7×M4(2) C2×D28 C4○D28 C28 C2×C14 M4(2) C8 C2×C4 C4 C22 C7 C1 # reps 1 2 2 1 1 1 1 1 3 6 3 6 6 1 6

Matrix representation of C8⋊D14 in GL4(𝔽113) generated by

 112 0 42 42 0 112 62 104 47 11 1 0 1 74 0 1
,
 103 25 0 0 99 1 0 0 0 7 9 88 91 48 104 0
,
 89 34 0 0 13 24 0 0 81 59 58 67 79 111 51 55
`G:=sub<GL(4,GF(113))| [112,0,47,1,0,112,11,74,42,62,1,0,42,104,0,1],[103,99,0,91,25,1,7,48,0,0,9,104,0,0,88,0],[89,13,81,79,34,24,59,111,0,0,58,51,0,0,67,55] >;`

C8⋊D14 in GAP, Magma, Sage, TeX

`C_8\rtimes D_{14}`
`% in TeX`

`G:=Group("C8:D14");`
`// GroupNames label`

`G:=SmallGroup(224,103);`
`// by ID`

`G=gap.SmallGroup(224,103);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,188,50,579,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^14=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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