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## G = D4⋊D14order 224 = 25·7

### 2nd semidirect product of D4 and D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D4⋊D14
 Chief series C1 — C7 — C14 — C28 — D28 — C2×D28 — D4⋊D14
 Lower central C7 — C14 — C28 — D4⋊D14
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for D4⋊D14
G = < a,b,c,d | a4=b2=c14=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >

Subgroups: 326 in 68 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, M4(2), D8, SD16, C2×D4, C4○D4, C28, C28, D14, C2×C14, C2×C14, C8⋊C22, C7⋊C8, D28, D28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C4.Dic7, D4⋊D7, Q8⋊D7, C2×D28, C7×C4○D4, D4⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8⋊C22, C7⋊D4, C22×D7, C2×C7⋊D4, D4⋊D14

Smallest permutation representation of D4⋊D14
On 56 points
Generators in S56
```(1 22 14 20)(2 23 8 21)(3 24 9 15)(4 25 10 16)(5 26 11 17)(6 27 12 18)(7 28 13 19)(29 56 36 49)(30 43 37 50)(31 44 38 51)(32 45 39 52)(33 46 40 53)(34 47 41 54)(35 48 42 55)
(1 54)(2 48)(3 56)(4 50)(5 44)(6 52)(7 46)(8 55)(9 49)(10 43)(11 51)(12 45)(13 53)(14 47)(15 36)(16 30)(17 38)(18 32)(19 40)(20 34)(21 42)(22 41)(23 35)(24 29)(25 37)(26 31)(27 39)(28 33)
(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 17)(2 16)(3 15)(4 21)(5 20)(6 19)(7 18)(8 25)(9 24)(10 23)(11 22)(12 28)(13 27)(14 26)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(43 48)(44 47)(45 46)(49 56)(50 55)(51 54)(52 53)```

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

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

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

41 conjugacy classes

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

41 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 D14 C7⋊D4 C7⋊D4 C8⋊C22 D4⋊D14 kernel D4⋊D14 C4.Dic7 D4⋊D7 Q8⋊D7 C2×D28 C7×C4○D4 C28 C2×C14 C4○D4 C2×C4 D4 Q8 C4 C22 C7 C1 # reps 1 1 2 2 1 1 1 1 3 3 3 3 6 6 1 6

Matrix representation of D4⋊D14 in GL4(𝔽113) generated by

 58 75 0 0 38 55 0 0 17 23 66 38 94 30 37 47
,
 82 36 8 43 87 46 65 106 4 30 112 5 46 108 33 99
,
 80 80 0 0 33 9 0 0 103 111 105 33 67 53 47 32
,
 80 80 0 0 9 33 0 0 29 87 68 58 82 91 82 45
`G:=sub<GL(4,GF(113))| [58,38,17,94,75,55,23,30,0,0,66,37,0,0,38,47],[82,87,4,46,36,46,30,108,8,65,112,33,43,106,5,99],[80,33,103,67,80,9,111,53,0,0,105,47,0,0,33,32],[80,9,29,82,80,33,87,91,0,0,68,82,0,0,58,45] >;`

D4⋊D14 in GAP, Magma, Sage, TeX

`D_4\rtimes D_{14}`
`% in TeX`

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

`G:=SmallGroup(224,144);`
`// by ID`

`G=gap.SmallGroup(224,144);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,188,579,159,69,6917]);`
`// 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,d*b*d=a^-1*b,d*c*d=c^-1>;`
`// generators/relations`

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