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## G = Q16⋊D14order 448 = 26·7

### 2nd semidirect product of Q16 and D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — Q16⋊D14
 Chief series C1 — C7 — C14 — C28 — C56 — D56 — C2×D56 — Q16⋊D14
 Lower central C7 — C14 — C28 — C56 — Q16⋊D14
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4○D8

Generators and relations for Q16⋊D14
G = < a,b,c,d | a8=c14=d2=1, b2=a4, bab-1=dad=a-1, ac=ca, cbc-1=a4b, dbd=a3b, dcd=c-1 >

Subgroups: 628 in 90 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, C28, C28, D14, C2×C14, C2×C14, M5(2), D16, SD32, C2×D8, C4○D8, C56, D28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C16⋊C22, C7⋊C16, D56, D56, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×D28, C7×C4○D4, C28.C8, C7⋊D16, C7⋊SD32, C2×D56, C7×C4○D8, Q16⋊D14
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C7⋊D4, C22×D7, C16⋊C22, D4⋊D7, C2×C7⋊D4, C2×D4⋊D7, Q16⋊D14

Smallest permutation representation of Q16⋊D14
On 112 points
Generators in S112
```(1 47 19 40 13 54 26 33)(2 48 20 41 14 55 27 34)(3 49 21 42 8 56 28 35)(4 43 15 36 9 50 22 29)(5 44 16 37 10 51 23 30)(6 45 17 38 11 52 24 31)(7 46 18 39 12 53 25 32)(57 96 79 112 64 89 72 105)(58 97 80 99 65 90 73 106)(59 98 81 100 66 91 74 107)(60 85 82 101 67 92 75 108)(61 86 83 102 68 93 76 109)(62 87 84 103 69 94 77 110)(63 88 71 104 70 95 78 111)
(1 60 13 67)(2 68 14 61)(3 62 8 69)(4 70 9 63)(5 64 10 57)(6 58 11 65)(7 66 12 59)(15 71 22 78)(16 79 23 72)(17 73 24 80)(18 81 25 74)(19 75 26 82)(20 83 27 76)(21 77 28 84)(29 95 36 88)(30 89 37 96)(31 97 38 90)(32 91 39 98)(33 85 40 92)(34 93 41 86)(35 87 42 94)(43 104 50 111)(44 112 51 105)(45 106 52 99)(46 100 53 107)(47 108 54 101)(48 102 55 109)(49 110 56 103)
(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)(57 58 59 60 61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 32)(2 31)(3 30)(4 29)(5 35)(6 34)(7 33)(8 37)(9 36)(10 42)(11 41)(12 40)(13 39)(14 38)(15 50)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 43)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(99 109)(100 108)(101 107)(102 106)(103 105)(110 112)```

`G:=sub<Sym(112)| (1,47,19,40,13,54,26,33)(2,48,20,41,14,55,27,34)(3,49,21,42,8,56,28,35)(4,43,15,36,9,50,22,29)(5,44,16,37,10,51,23,30)(6,45,17,38,11,52,24,31)(7,46,18,39,12,53,25,32)(57,96,79,112,64,89,72,105)(58,97,80,99,65,90,73,106)(59,98,81,100,66,91,74,107)(60,85,82,101,67,92,75,108)(61,86,83,102,68,93,76,109)(62,87,84,103,69,94,77,110)(63,88,71,104,70,95,78,111), (1,60,13,67)(2,68,14,61)(3,62,8,69)(4,70,9,63)(5,64,10,57)(6,58,11,65)(7,66,12,59)(15,71,22,78)(16,79,23,72)(17,73,24,80)(18,81,25,74)(19,75,26,82)(20,83,27,76)(21,77,28,84)(29,95,36,88)(30,89,37,96)(31,97,38,90)(32,91,39,98)(33,85,40,92)(34,93,41,86)(35,87,42,94)(43,104,50,111)(44,112,51,105)(45,106,52,99)(46,100,53,107)(47,108,54,101)(48,102,55,109)(49,110,56,103), (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,32)(2,31)(3,30)(4,29)(5,35)(6,34)(7,33)(8,37)(9,36)(10,42)(11,41)(12,40)(13,39)(14,38)(15,50)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,43)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112)>;`

`G:=Group( (1,47,19,40,13,54,26,33)(2,48,20,41,14,55,27,34)(3,49,21,42,8,56,28,35)(4,43,15,36,9,50,22,29)(5,44,16,37,10,51,23,30)(6,45,17,38,11,52,24,31)(7,46,18,39,12,53,25,32)(57,96,79,112,64,89,72,105)(58,97,80,99,65,90,73,106)(59,98,81,100,66,91,74,107)(60,85,82,101,67,92,75,108)(61,86,83,102,68,93,76,109)(62,87,84,103,69,94,77,110)(63,88,71,104,70,95,78,111), (1,60,13,67)(2,68,14,61)(3,62,8,69)(4,70,9,63)(5,64,10,57)(6,58,11,65)(7,66,12,59)(15,71,22,78)(16,79,23,72)(17,73,24,80)(18,81,25,74)(19,75,26,82)(20,83,27,76)(21,77,28,84)(29,95,36,88)(30,89,37,96)(31,97,38,90)(32,91,39,98)(33,85,40,92)(34,93,41,86)(35,87,42,94)(43,104,50,111)(44,112,51,105)(45,106,52,99)(46,100,53,107)(47,108,54,101)(48,102,55,109)(49,110,56,103), (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,32)(2,31)(3,30)(4,29)(5,35)(6,34)(7,33)(8,37)(9,36)(10,42)(11,41)(12,40)(13,39)(14,38)(15,50)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,43)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112) );`

`G=PermutationGroup([[(1,47,19,40,13,54,26,33),(2,48,20,41,14,55,27,34),(3,49,21,42,8,56,28,35),(4,43,15,36,9,50,22,29),(5,44,16,37,10,51,23,30),(6,45,17,38,11,52,24,31),(7,46,18,39,12,53,25,32),(57,96,79,112,64,89,72,105),(58,97,80,99,65,90,73,106),(59,98,81,100,66,91,74,107),(60,85,82,101,67,92,75,108),(61,86,83,102,68,93,76,109),(62,87,84,103,69,94,77,110),(63,88,71,104,70,95,78,111)], [(1,60,13,67),(2,68,14,61),(3,62,8,69),(4,70,9,63),(5,64,10,57),(6,58,11,65),(7,66,12,59),(15,71,22,78),(16,79,23,72),(17,73,24,80),(18,81,25,74),(19,75,26,82),(20,83,27,76),(21,77,28,84),(29,95,36,88),(30,89,37,96),(31,97,38,90),(32,91,39,98),(33,85,40,92),(34,93,41,86),(35,87,42,94),(43,104,50,111),(44,112,51,105),(45,106,52,99),(46,100,53,107),(47,108,54,101),(48,102,55,109),(49,110,56,103)], [(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),(57,58,59,60,61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,32),(2,31),(3,30),(4,29),(5,35),(6,34),(7,33),(8,37),(9,36),(10,42),(11,41),(12,40),(13,39),(14,38),(15,50),(16,56),(17,55),(18,54),(19,53),(20,52),(21,51),(22,43),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(57,84),(58,83),(59,82),(60,81),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(99,109),(100,108),(101,107),(102,106),(103,105),(110,112)]])`

58 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 7A 7B 7C 8A 8B 8C 14A 14B 14C 14D 14E 14F 14G ··· 14L 16A 16B 16C 16D 28A ··· 28F 28G 28H 28I 28J ··· 28O 56A ··· 56L order 1 2 2 2 2 2 4 4 4 7 7 7 8 8 8 14 14 14 14 14 14 14 ··· 14 16 16 16 16 28 ··· 28 28 28 28 28 ··· 28 56 ··· 56 size 1 1 2 8 56 56 2 2 8 2 2 2 2 2 4 2 2 2 4 4 4 8 ··· 8 28 28 28 28 2 ··· 2 4 4 4 8 ··· 8 4 ··· 4

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 D14 C7⋊D4 C7⋊D4 C16⋊C22 D4⋊D7 D4⋊D7 Q16⋊D14 kernel Q16⋊D14 C28.C8 C7⋊D16 C7⋊SD32 C2×D56 C7×C4○D8 C56 C2×C28 C4○D8 C28 C2×C14 C2×C8 D8 Q16 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 2 2 1 1 1 1 3 2 2 3 3 3 6 6 2 3 3 12

Matrix representation of Q16⋊D14 in GL4(𝔽113) generated by

 41 48 0 0 65 21 0 0 11 61 21 65 52 108 48 41
,
 14 62 2 0 51 7 0 2 72 27 99 51 86 89 62 106
,
 33 33 0 0 80 104 0 0 2 96 80 80 17 75 33 9
,
 108 17 0 0 85 5 0 0 38 62 4 109 65 75 32 109
`G:=sub<GL(4,GF(113))| [41,65,11,52,48,21,61,108,0,0,21,48,0,0,65,41],[14,51,72,86,62,7,27,89,2,0,99,62,0,2,51,106],[33,80,2,17,33,104,96,75,0,0,80,33,0,0,80,9],[108,85,38,65,17,5,62,75,0,0,4,32,0,0,109,109] >;`

Q16⋊D14 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes D_{14}`
`% in TeX`

`G:=Group("Q16:D14");`
`// GroupNames label`

`G:=SmallGroup(448,727);`
`// by ID`

`G=gap.SmallGroup(448,727);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,387,675,185,192,1684,438,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^14=d^2=1,b^2=a^4,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

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