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## G = Q8⋊3D21order 336 = 24·3·7

### The semidirect product of Q8 and D21 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — Q8⋊3D21
 Chief series C1 — C7 — C21 — C42 — D42 — C4×D21 — Q8⋊3D21
 Lower central C21 — C42 — Q8⋊3D21
 Upper central C1 — C2 — Q8

Generators and relations for Q83D21
G = < a,b,c,d | a4=c21=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 568 in 80 conjugacy classes, 35 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C7, C2×C4, D4, Q8, Dic3, C12, D6, D7, C14, C4○D4, C21, C4×S3, D12, C3×Q8, Dic7, C28, D14, D21, C42, Q83S3, C4×D7, D28, C7×Q8, Dic21, C84, D42, Q82D7, C4×D21, D84, Q8×C21, Q83D21
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, D21, Q83S3, C22×D7, D42, Q82D7, C22×D21, Q83D21

Smallest permutation representation of Q83D21
On 168 points
Generators in S168
```(1 71 39 49)(2 72 40 50)(3 73 41 51)(4 74 42 52)(5 75 22 53)(6 76 23 54)(7 77 24 55)(8 78 25 56)(9 79 26 57)(10 80 27 58)(11 81 28 59)(12 82 29 60)(13 83 30 61)(14 84 31 62)(15 64 32 63)(16 65 33 43)(17 66 34 44)(18 67 35 45)(19 68 36 46)(20 69 37 47)(21 70 38 48)(85 132 118 153)(86 133 119 154)(87 134 120 155)(88 135 121 156)(89 136 122 157)(90 137 123 158)(91 138 124 159)(92 139 125 160)(93 140 126 161)(94 141 106 162)(95 142 107 163)(96 143 108 164)(97 144 109 165)(98 145 110 166)(99 146 111 167)(100 147 112 168)(101 127 113 148)(102 128 114 149)(103 129 115 150)(104 130 116 151)(105 131 117 152)
(1 123 39 90)(2 124 40 91)(3 125 41 92)(4 126 42 93)(5 106 22 94)(6 107 23 95)(7 108 24 96)(8 109 25 97)(9 110 26 98)(10 111 27 99)(11 112 28 100)(12 113 29 101)(13 114 30 102)(14 115 31 103)(15 116 32 104)(16 117 33 105)(17 118 34 85)(18 119 35 86)(19 120 36 87)(20 121 37 88)(21 122 38 89)(43 152 65 131)(44 153 66 132)(45 154 67 133)(46 155 68 134)(47 156 69 135)(48 157 70 136)(49 158 71 137)(50 159 72 138)(51 160 73 139)(52 161 74 140)(53 162 75 141)(54 163 76 142)(55 164 77 143)(56 165 78 144)(57 166 79 145)(58 167 80 146)(59 168 81 147)(60 148 82 127)(61 149 83 128)(62 150 84 129)(63 151 64 130)
(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)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147)(148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 66)(23 65)(24 64)(25 84)(26 83)(27 82)(28 81)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 70)(40 69)(41 68)(42 67)(85 141)(86 140)(87 139)(88 138)(89 137)(90 136)(91 135)(92 134)(93 133)(94 132)(95 131)(96 130)(97 129)(98 128)(99 127)(100 147)(101 146)(102 145)(103 144)(104 143)(105 142)(106 153)(107 152)(108 151)(109 150)(110 149)(111 148)(112 168)(113 167)(114 166)(115 165)(116 164)(117 163)(118 162)(119 161)(120 160)(121 159)(122 158)(123 157)(124 156)(125 155)(126 154)```

`G:=sub<Sym(168)| (1,71,39,49)(2,72,40,50)(3,73,41,51)(4,74,42,52)(5,75,22,53)(6,76,23,54)(7,77,24,55)(8,78,25,56)(9,79,26,57)(10,80,27,58)(11,81,28,59)(12,82,29,60)(13,83,30,61)(14,84,31,62)(15,64,32,63)(16,65,33,43)(17,66,34,44)(18,67,35,45)(19,68,36,46)(20,69,37,47)(21,70,38,48)(85,132,118,153)(86,133,119,154)(87,134,120,155)(88,135,121,156)(89,136,122,157)(90,137,123,158)(91,138,124,159)(92,139,125,160)(93,140,126,161)(94,141,106,162)(95,142,107,163)(96,143,108,164)(97,144,109,165)(98,145,110,166)(99,146,111,167)(100,147,112,168)(101,127,113,148)(102,128,114,149)(103,129,115,150)(104,130,116,151)(105,131,117,152), (1,123,39,90)(2,124,40,91)(3,125,41,92)(4,126,42,93)(5,106,22,94)(6,107,23,95)(7,108,24,96)(8,109,25,97)(9,110,26,98)(10,111,27,99)(11,112,28,100)(12,113,29,101)(13,114,30,102)(14,115,31,103)(15,116,32,104)(16,117,33,105)(17,118,34,85)(18,119,35,86)(19,120,36,87)(20,121,37,88)(21,122,38,89)(43,152,65,131)(44,153,66,132)(45,154,67,133)(46,155,68,134)(47,156,69,135)(48,157,70,136)(49,158,71,137)(50,159,72,138)(51,160,73,139)(52,161,74,140)(53,162,75,141)(54,163,76,142)(55,164,77,143)(56,165,78,144)(57,166,79,145)(58,167,80,146)(59,168,81,147)(60,148,82,127)(61,149,83,128)(62,150,84,129)(63,151,64,130), (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,66)(23,65)(24,64)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(85,141)(86,140)(87,139)(88,138)(89,137)(90,136)(91,135)(92,134)(93,133)(94,132)(95,131)(96,130)(97,129)(98,128)(99,127)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,168)(113,167)(114,166)(115,165)(116,164)(117,163)(118,162)(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)>;`

`G:=Group( (1,71,39,49)(2,72,40,50)(3,73,41,51)(4,74,42,52)(5,75,22,53)(6,76,23,54)(7,77,24,55)(8,78,25,56)(9,79,26,57)(10,80,27,58)(11,81,28,59)(12,82,29,60)(13,83,30,61)(14,84,31,62)(15,64,32,63)(16,65,33,43)(17,66,34,44)(18,67,35,45)(19,68,36,46)(20,69,37,47)(21,70,38,48)(85,132,118,153)(86,133,119,154)(87,134,120,155)(88,135,121,156)(89,136,122,157)(90,137,123,158)(91,138,124,159)(92,139,125,160)(93,140,126,161)(94,141,106,162)(95,142,107,163)(96,143,108,164)(97,144,109,165)(98,145,110,166)(99,146,111,167)(100,147,112,168)(101,127,113,148)(102,128,114,149)(103,129,115,150)(104,130,116,151)(105,131,117,152), (1,123,39,90)(2,124,40,91)(3,125,41,92)(4,126,42,93)(5,106,22,94)(6,107,23,95)(7,108,24,96)(8,109,25,97)(9,110,26,98)(10,111,27,99)(11,112,28,100)(12,113,29,101)(13,114,30,102)(14,115,31,103)(15,116,32,104)(16,117,33,105)(17,118,34,85)(18,119,35,86)(19,120,36,87)(20,121,37,88)(21,122,38,89)(43,152,65,131)(44,153,66,132)(45,154,67,133)(46,155,68,134)(47,156,69,135)(48,157,70,136)(49,158,71,137)(50,159,72,138)(51,160,73,139)(52,161,74,140)(53,162,75,141)(54,163,76,142)(55,164,77,143)(56,165,78,144)(57,166,79,145)(58,167,80,146)(59,168,81,147)(60,148,82,127)(61,149,83,128)(62,150,84,129)(63,151,64,130), (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,66)(23,65)(24,64)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(85,141)(86,140)(87,139)(88,138)(89,137)(90,136)(91,135)(92,134)(93,133)(94,132)(95,131)(96,130)(97,129)(98,128)(99,127)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,168)(113,167)(114,166)(115,165)(116,164)(117,163)(118,162)(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154) );`

`G=PermutationGroup([[(1,71,39,49),(2,72,40,50),(3,73,41,51),(4,74,42,52),(5,75,22,53),(6,76,23,54),(7,77,24,55),(8,78,25,56),(9,79,26,57),(10,80,27,58),(11,81,28,59),(12,82,29,60),(13,83,30,61),(14,84,31,62),(15,64,32,63),(16,65,33,43),(17,66,34,44),(18,67,35,45),(19,68,36,46),(20,69,37,47),(21,70,38,48),(85,132,118,153),(86,133,119,154),(87,134,120,155),(88,135,121,156),(89,136,122,157),(90,137,123,158),(91,138,124,159),(92,139,125,160),(93,140,126,161),(94,141,106,162),(95,142,107,163),(96,143,108,164),(97,144,109,165),(98,145,110,166),(99,146,111,167),(100,147,112,168),(101,127,113,148),(102,128,114,149),(103,129,115,150),(104,130,116,151),(105,131,117,152)], [(1,123,39,90),(2,124,40,91),(3,125,41,92),(4,126,42,93),(5,106,22,94),(6,107,23,95),(7,108,24,96),(8,109,25,97),(9,110,26,98),(10,111,27,99),(11,112,28,100),(12,113,29,101),(13,114,30,102),(14,115,31,103),(15,116,32,104),(16,117,33,105),(17,118,34,85),(18,119,35,86),(19,120,36,87),(20,121,37,88),(21,122,38,89),(43,152,65,131),(44,153,66,132),(45,154,67,133),(46,155,68,134),(47,156,69,135),(48,157,70,136),(49,158,71,137),(50,159,72,138),(51,160,73,139),(52,161,74,140),(53,162,75,141),(54,163,76,142),(55,164,77,143),(56,165,78,144),(57,166,79,145),(58,167,80,146),(59,168,81,147),(60,148,82,127),(61,149,83,128),(62,150,84,129),(63,151,64,130)], [(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),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147),(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,63),(8,62),(9,61),(10,60),(11,59),(12,58),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,66),(23,65),(24,64),(25,84),(26,83),(27,82),(28,81),(29,80),(30,79),(31,78),(32,77),(33,76),(34,75),(35,74),(36,73),(37,72),(38,71),(39,70),(40,69),(41,68),(42,67),(85,141),(86,140),(87,139),(88,138),(89,137),(90,136),(91,135),(92,134),(93,133),(94,132),(95,131),(96,130),(97,129),(98,128),(99,127),(100,147),(101,146),(102,145),(103,144),(104,143),(105,142),(106,153),(107,152),(108,151),(109,150),(110,149),(111,148),(112,168),(113,167),(114,166),(115,165),(116,164),(117,163),(118,162),(119,161),(120,160),(121,159),(122,158),(123,157),(124,156),(125,155),(126,154)]])`

60 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6 7A 7B 7C 12A 12B 12C 14A 14B 14C 21A ··· 21F 28A ··· 28I 42A ··· 42F 84A ··· 84R order 1 2 2 2 2 3 4 4 4 4 4 6 7 7 7 12 12 12 14 14 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 42 42 42 2 2 2 2 21 21 2 2 2 2 4 4 4 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D6 D7 C4○D4 D14 D21 D42 Q8⋊3S3 Q8⋊2D7 Q8⋊3D21 kernel Q8⋊3D21 C4×D21 D84 Q8×C21 C7×Q8 C28 C3×Q8 C21 C12 Q8 C4 C7 C3 C1 # reps 1 3 3 1 1 3 3 2 9 6 18 1 3 6

Matrix representation of Q83D21 in GL4(𝔽337) generated by

 336 0 0 0 0 336 0 0 0 0 1 313 0 0 309 336
,
 1 0 0 0 0 1 0 0 0 0 148 155 0 0 0 189
,
 253 30 0 0 116 107 0 0 0 0 1 0 0 0 0 1
,
 230 64 0 0 116 107 0 0 0 0 1 313 0 0 0 336
`G:=sub<GL(4,GF(337))| [336,0,0,0,0,336,0,0,0,0,1,309,0,0,313,336],[1,0,0,0,0,1,0,0,0,0,148,0,0,0,155,189],[253,116,0,0,30,107,0,0,0,0,1,0,0,0,0,1],[230,116,0,0,64,107,0,0,0,0,1,0,0,0,313,336] >;`

Q83D21 in GAP, Magma, Sage, TeX

`Q_8\rtimes_3D_{21}`
`% in TeX`

`G:=Group("Q8:3D21");`
`// GroupNames label`

`G:=SmallGroup(336,201);`
`// by ID`

`G=gap.SmallGroup(336,201);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,218,116,50,964,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^21=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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