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## G = D4.D21order 336 = 24·3·7

### The non-split extension by D4 of D21 acting via D21/C21=C2

Aliases: D4.D21, C4.2D42, C28.10D6, C42.35D4, C2110SD16, Dic422C2, C12.10D14, C84.2C22, C21⋊C82C2, C33(D4.D7), C73(D4.S3), (C7×D4).1S3, (C3×D4).1D7, (D4×C21).1C2, C6.17(C7⋊D4), C2.5(C217D4), C14.17(C3⋊D4), SmallGroup(336,102)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C84 — D4.D21
 Chief series C1 — C7 — C21 — C42 — C84 — Dic42 — D4.D21
 Lower central C21 — C42 — C84 — D4.D21
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4.D21
G = < a,b,c,d | a4=b2=c21=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Smallest permutation representation of D4.D21
On 168 points
Generators in S168
```(1 72 29 56)(2 73 30 57)(3 74 31 58)(4 75 32 59)(5 76 33 60)(6 77 34 61)(7 78 35 62)(8 79 36 63)(9 80 37 43)(10 81 38 44)(11 82 39 45)(12 83 40 46)(13 84 41 47)(14 64 42 48)(15 65 22 49)(16 66 23 50)(17 67 24 51)(18 68 25 52)(19 69 26 53)(20 70 27 54)(21 71 28 55)(85 127 121 165)(86 128 122 166)(87 129 123 167)(88 130 124 168)(89 131 125 148)(90 132 126 149)(91 133 106 150)(92 134 107 151)(93 135 108 152)(94 136 109 153)(95 137 110 154)(96 138 111 155)(97 139 112 156)(98 140 113 157)(99 141 114 158)(100 142 115 159)(101 143 116 160)(102 144 117 161)(103 145 118 162)(104 146 119 163)(105 147 120 164)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 64)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
(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)
(1 118 29 103)(2 117 30 102)(3 116 31 101)(4 115 32 100)(5 114 33 99)(6 113 34 98)(7 112 35 97)(8 111 36 96)(9 110 37 95)(10 109 38 94)(11 108 39 93)(12 107 40 92)(13 106 41 91)(14 126 42 90)(15 125 22 89)(16 124 23 88)(17 123 24 87)(18 122 25 86)(19 121 26 85)(20 120 27 105)(21 119 28 104)(43 154 80 137)(44 153 81 136)(45 152 82 135)(46 151 83 134)(47 150 84 133)(48 149 64 132)(49 148 65 131)(50 168 66 130)(51 167 67 129)(52 166 68 128)(53 165 69 127)(54 164 70 147)(55 163 71 146)(56 162 72 145)(57 161 73 144)(58 160 74 143)(59 159 75 142)(60 158 76 141)(61 157 77 140)(62 156 78 139)(63 155 79 138)```

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

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

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

57 conjugacy classes

 class 1 2A 2B 3 4A 4B 6A 6B 6C 7A 7B 7C 8A 8B 12 14A 14B 14C 14D ··· 14I 21A ··· 21F 28A 28B 28C 42A ··· 42F 42G ··· 42R 84A ··· 84F order 1 2 2 3 4 4 6 6 6 7 7 7 8 8 12 14 14 14 14 ··· 14 21 ··· 21 28 28 28 42 ··· 42 42 ··· 42 84 ··· 84 size 1 1 4 2 2 84 2 4 4 2 2 2 42 42 4 2 2 2 4 ··· 4 2 ··· 2 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - - - image C1 C2 C2 C2 S3 D4 D6 D7 SD16 C3⋊D4 D14 D21 C7⋊D4 D42 C21⋊7D4 D4.S3 D4.D7 D4.D21 kernel D4.D21 C21⋊C8 Dic42 D4×C21 C7×D4 C42 C28 C3×D4 C21 C14 C12 D4 C6 C4 C2 C7 C3 C1 # reps 1 1 1 1 1 1 1 3 2 2 3 6 6 6 12 1 3 6

Matrix representation of D4.D21 in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 336 252 0 0 115 1
,
 1 0 0 0 0 1 0 0 0 0 336 252 0 0 0 1
,
 32 0 0 0 188 158 0 0 0 0 1 0 0 0 0 1
,
 48 25 0 0 164 289 0 0 0 0 141 95 0 0 149 196
`G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,336,115,0,0,252,1],[1,0,0,0,0,1,0,0,0,0,336,0,0,0,252,1],[32,188,0,0,0,158,0,0,0,0,1,0,0,0,0,1],[48,164,0,0,25,289,0,0,0,0,141,149,0,0,95,196] >;`

D4.D21 in GAP, Magma, Sage, TeX

`D_4.D_{21}`
`% in TeX`

`G:=Group("D4.D21");`
`// GroupNames label`

`G:=SmallGroup(336,102);`
`// by ID`

`G=gap.SmallGroup(336,102);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,73,218,116,50,964,10373]);`
`// Polycyclic`

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

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