metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊3S3, Dic6⋊3D7, D14.2D6, C28.14D6, C12.14D14, C42.3C23, C84.26C22, Dic3.2D14, D42.8C22, Dic21.10C22, (C3×D28)⋊5C2, (C4×D21)⋊5C2, C21⋊2(C4○D4), C3⋊D28⋊1C2, C4.19(S3×D7), C7⋊1(D4⋊2S3), (Dic3×D7)⋊2C2, (C7×Dic6)⋊5C2, C3⋊2(Q8⋊2D7), C6.3(C22×D7), C14.3(C22×S3), (C6×D7).2C22, (C7×Dic3).2C22, C2.7(C2×S3×D7), SmallGroup(336,139)
Series: Derived ►Chief ►Lower central ►Upper central
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, D4⋊2S3, C4×D7, D28, D28, C7×Q8, C7×Dic3, Dic21, C84, C6×D7, D42, Q8⋊2D7, 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, D4⋊2S3, C22×D7, S3×D7, Q8⋊2D7, C2×S3×D7, 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