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## G = D28⋊S3order 336 = 24·3·7

### 3rd semidirect product of D28 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D28⋊S3
 Chief series C1 — C7 — C21 — C42 — C6×D7 — Dic3×D7 — D28⋊S3
 Lower central C21 — C42 — D28⋊S3
 Upper central C1 — C2 — C4

Generators and relations for D28⋊S3
G = < a,b,c,d | a28=b2=c3=d2=1, bab=a-1, ac=ca, dad=a13, bc=cb, dbd=a26b, dcd=c-1 >

Subgroups: 468 in 80 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C7, C2×C4, D4, Q8, Dic3, Dic3, C12, D6, C2×C6, D7, C14, C4○D4, C21, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, Dic7, C28, C28, D14, D14, C3×D7, D21, C42, D42S3, C4×D7, D28, D28, C7×Q8, C7×Dic3, Dic21, C84, C6×D7, D42, Q82D7, Dic3×D7, C3⋊D28, C3×D28, C7×Dic6, C4×D21, D28⋊S3
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, D42S3, C22×D7, S3×D7, Q82D7, C2×S3×D7, D28⋊S3

Smallest permutation representation of D28⋊S3
On 168 points
Generators in S168
```(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)
(1 159)(2 158)(3 157)(4 156)(5 155)(6 154)(7 153)(8 152)(9 151)(10 150)(11 149)(12 148)(13 147)(14 146)(15 145)(16 144)(17 143)(18 142)(19 141)(20 168)(21 167)(22 166)(23 165)(24 164)(25 163)(26 162)(27 161)(28 160)(29 136)(30 135)(31 134)(32 133)(33 132)(34 131)(35 130)(36 129)(37 128)(38 127)(39 126)(40 125)(41 124)(42 123)(43 122)(44 121)(45 120)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 140)(54 139)(55 138)(56 137)(57 109)(58 108)(59 107)(60 106)(61 105)(62 104)(63 103)(64 102)(65 101)(66 100)(67 99)(68 98)(69 97)(70 96)(71 95)(72 94)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 112)(83 111)(84 110)
(1 62 41)(2 63 42)(3 64 43)(4 65 44)(5 66 45)(6 67 46)(7 68 47)(8 69 48)(9 70 49)(10 71 50)(11 72 51)(12 73 52)(13 74 53)(14 75 54)(15 76 55)(16 77 56)(17 78 29)(18 79 30)(19 80 31)(20 81 32)(21 82 33)(22 83 34)(23 84 35)(24 57 36)(25 58 37)(26 59 38)(27 60 39)(28 61 40)(85 133 168)(86 134 141)(87 135 142)(88 136 143)(89 137 144)(90 138 145)(91 139 146)(92 140 147)(93 113 148)(94 114 149)(95 115 150)(96 116 151)(97 117 152)(98 118 153)(99 119 154)(100 120 155)(101 121 156)(102 122 157)(103 123 158)(104 124 159)(105 125 160)(106 126 161)(107 127 162)(108 128 163)(109 129 164)(110 130 165)(111 131 166)(112 132 167)
(2 14)(3 27)(4 12)(5 25)(6 10)(7 23)(9 21)(11 19)(13 17)(16 28)(18 26)(20 24)(29 74)(30 59)(31 72)(32 57)(33 70)(34 83)(35 68)(36 81)(37 66)(38 79)(39 64)(40 77)(41 62)(42 75)(43 60)(44 73)(45 58)(46 71)(47 84)(48 69)(49 82)(50 67)(51 80)(52 65)(53 78)(54 63)(55 76)(56 61)(85 131)(86 116)(87 129)(88 114)(89 127)(90 140)(91 125)(92 138)(93 123)(94 136)(95 121)(96 134)(97 119)(98 132)(99 117)(100 130)(101 115)(102 128)(103 113)(104 126)(105 139)(106 124)(107 137)(108 122)(109 135)(110 120)(111 133)(112 118)(141 151)(142 164)(143 149)(144 162)(145 147)(146 160)(148 158)(150 156)(152 154)(153 167)(155 165)(157 163)(159 161)(166 168)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 D7 C4○D4 D14 D14 D4⋊2S3 S3×D7 Q8⋊2D7 C2×S3×D7 D28⋊S3 kernel D28⋊S3 Dic3×D7 C3⋊D28 C3×D28 C7×Dic6 C4×D21 D28 C28 D14 Dic6 C21 Dic3 C12 C7 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 1 2 3 2 6 3 1 3 3 3 6

Matrix representation of D28⋊S3 in GL6(𝔽337)

 336 0 0 0 0 0 0 336 0 0 0 0 0 0 194 110 0 0 0 0 85 109 0 0 0 0 0 0 189 0 0 0 0 0 120 148
,
 336 0 0 0 0 0 0 336 0 0 0 0 0 0 0 33 0 0 0 0 143 0 0 0 0 0 0 0 320 3 0 0 0 0 241 17
,
 336 336 0 0 0 0 1 0 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 0 336 336 0 0 0 0 0 0 109 1 0 0 0 0 252 228 0 0 0 0 0 0 1 0 0 0 0 0 236 336

`G:=sub<GL(6,GF(337))| [336,0,0,0,0,0,0,336,0,0,0,0,0,0,194,85,0,0,0,0,110,109,0,0,0,0,0,0,189,120,0,0,0,0,0,148],[336,0,0,0,0,0,0,336,0,0,0,0,0,0,0,143,0,0,0,0,33,0,0,0,0,0,0,0,320,241,0,0,0,0,3,17],[336,1,0,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,336,0,0,0,0,0,336,0,0,0,0,0,0,109,252,0,0,0,0,1,228,0,0,0,0,0,0,1,236,0,0,0,0,0,336] >;`

D28⋊S3 in GAP, Magma, Sage, TeX

`D_{28}\rtimes S_3`
`% in TeX`

`G:=Group("D28:S3");`
`// GroupNames label`

`G:=SmallGroup(336,139);`
`// by ID`

`G=gap.SmallGroup(336,139);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,218,116,50,490,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^13,b*c=c*b,d*b*d=a^26*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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