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## G = Q16⋊D7order 224 = 25·7

### 2nd semidirect product of Q16 and D7 acting via D7/C7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — Q16⋊D7
 Chief series C1 — C7 — C14 — C28 — C4×D7 — Q8×D7 — Q16⋊D7
 Lower central C7 — C14 — C28 — Q16⋊D7
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q16⋊D7
G = < a,b,c,d | a8=c7=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 270 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, Q8, D7, C14, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C8.C22, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C7×Q8, C8⋊D7, C56⋊C2, Q8⋊D7, C7⋊Q16, C7×Q16, Q8×D7, Q82D7, Q16⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8.C22, C22×D7, D4×D7, Q16⋊D7

Smallest permutation representation of Q16⋊D7
On 112 points
Generators in S112
```(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 48 13 44)(10 47 14 43)(11 46 15 42)(12 45 16 41)(17 61 21 57)(18 60 22 64)(19 59 23 63)(20 58 24 62)(33 83 37 87)(34 82 38 86)(35 81 39 85)(36 88 40 84)(49 110 53 106)(50 109 54 105)(51 108 55 112)(52 107 56 111)(65 94 69 90)(66 93 70 89)(67 92 71 96)(68 91 72 95)(73 99 77 103)(74 98 78 102)(75 97 79 101)(76 104 80 100)
(1 102 14 89 107 61 36)(2 103 15 90 108 62 37)(3 104 16 91 109 63 38)(4 97 9 92 110 64 39)(5 98 10 93 111 57 40)(6 99 11 94 112 58 33)(7 100 12 95 105 59 34)(8 101 13 96 106 60 35)(17 84 29 78 47 70 52)(18 85 30 79 48 71 53)(19 86 31 80 41 72 54)(20 87 32 73 42 65 55)(21 88 25 74 43 66 56)(22 81 26 75 44 67 49)(23 82 27 76 45 68 50)(24 83 28 77 46 69 51)
(1 84)(2 81)(3 86)(4 83)(5 88)(6 85)(7 82)(8 87)(9 51)(10 56)(11 53)(12 50)(13 55)(14 52)(15 49)(16 54)(17 102)(18 99)(19 104)(20 101)(21 98)(22 103)(23 100)(24 97)(25 40)(26 37)(27 34)(28 39)(29 36)(30 33)(31 38)(32 35)(41 109)(42 106)(43 111)(44 108)(45 105)(46 110)(47 107)(48 112)(57 74)(58 79)(59 76)(60 73)(61 78)(62 75)(63 80)(64 77)(65 96)(66 93)(67 90)(68 95)(69 92)(70 89)(71 94)(72 91)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 C8.C22 D4×D7 Q16⋊D7 kernel Q16⋊D7 C8⋊D7 C56⋊C2 Q8⋊D7 C7⋊Q16 C7×Q16 Q8×D7 Q8⋊2D7 Dic7 D14 Q16 C8 Q8 C7 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 3 3 6 1 3 6

Matrix representation of Q16⋊D7 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 82 48 41 41 0 0 4 0 0 4 0 0 109 37 72 37 0 0 42 37 72 72
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 36 0 0 0 0 0 112 1 0 0 69 0 1 0 0 0 69 112 1 0
,
 112 1 0 0 0 0 32 80 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 81 112 0 0 0 0 0 0 112 0 36 0 0 0 44 0 112 112 0 0 0 0 1 0 0 0 69 112 1 0

`G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,82,4,109,42,0,0,48,0,37,37,0,0,41,0,72,72,0,0,41,4,37,72],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,69,69,0,0,0,0,0,112,0,0,36,112,1,1,0,0,0,1,0,0],[112,32,0,0,0,0,1,80,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,81,0,0,0,0,0,112,0,0,0,0,0,0,112,44,0,69,0,0,0,0,0,112,0,0,36,112,1,1,0,0,0,112,0,0] >;`

Q16⋊D7 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes D_7`
`% in TeX`

`G:=Group("Q16:D7");`
`// GroupNames label`

`G:=SmallGroup(224,113);`
`// by ID`

`G=gap.SmallGroup(224,113);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,362,116,86,297,159,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^7=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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