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## G = D4⋊8D14order 224 = 25·7

### 4th semidirect product of D4 and D14 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D4⋊8D14
 Chief series C1 — C7 — C14 — D14 — C22×D7 — D4×D7 — D4⋊8D14
 Lower central C7 — C14 — D4⋊8D14
 Upper central C1 — C2 — C4○D4

Generators and relations for D48D14
G = < a,b,c,d | a4=b2=c14=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 742 in 166 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×D4, C4○D4, C4○D4, Dic7, C28, C28, D14, D14, C2×C14, 2+ 1+4, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, C2×D28, C4○D28, D4×D7, Q82D7, C7×C4○D4, D48D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C22×D7, C23×D7, D48D14

Smallest permutation representation of D48D14
On 56 points
Generators in S56
```(1 48 41 28)(2 49 42 15)(3 50 29 16)(4 51 30 17)(5 52 31 18)(6 53 32 19)(7 54 33 20)(8 55 34 21)(9 56 35 22)(10 43 36 23)(11 44 37 24)(12 45 38 25)(13 46 39 26)(14 47 40 27)
(1 28)(2 49)(3 16)(4 51)(5 18)(6 53)(7 20)(8 55)(9 22)(10 43)(11 24)(12 45)(13 26)(14 47)(15 42)(17 30)(19 32)(21 34)(23 36)(25 38)(27 40)(29 50)(31 52)(33 54)(35 56)(37 44)(39 46)(41 48)
(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 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 56)(40 55)(41 54)(42 53)```

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2J 4A 4B 4C 4D 4E 4F 7A 7B 7C 14A 14B 14C 14D ··· 14L 28A ··· 28F 28G ··· 28O order 1 2 2 2 2 2 ··· 2 4 4 4 4 4 4 7 7 7 14 14 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 2 2 14 ··· 14 2 2 2 2 14 14 2 2 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D7 D14 D14 D14 2+ 1+4 D4⋊8D14 kernel D4⋊8D14 C2×D28 C4○D28 D4×D7 Q8⋊2D7 C7×C4○D4 C4○D4 C2×C4 D4 Q8 C7 C1 # reps 1 3 3 6 2 1 3 9 9 3 1 6

Matrix representation of D48D14 in GL4(𝔽29) generated by

 27 6 6 23 4 2 0 25 0 0 8 23 0 0 6 21
,
 27 6 6 23 4 2 0 25 10 14 8 23 0 14 6 21
,
 26 19 1 21 3 0 7 7 0 0 22 10 0 0 19 10
,
 2 23 23 6 15 27 25 0 0 0 23 8 0 0 21 6
`G:=sub<GL(4,GF(29))| [27,4,0,0,6,2,0,0,6,0,8,6,23,25,23,21],[27,4,10,0,6,2,14,14,6,0,8,6,23,25,23,21],[26,3,0,0,19,0,0,0,1,7,22,19,21,7,10,10],[2,15,0,0,23,27,0,0,23,25,23,21,6,0,8,6] >;`

D48D14 in GAP, Magma, Sage, TeX

`D_4\rtimes_8D_{14}`
`% in TeX`

`G:=Group("D4:8D14");`
`// GroupNames label`

`G:=SmallGroup(224,185);`
`// by ID`

`G=gap.SmallGroup(224,185);`
`# by ID`

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

`G:=Group<a,b,c,d|a^4=b^2=c^14=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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