metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊Dic7, C42.14D4, C14.10D12, C7⋊3(D6⋊C4), (S3×C14)⋊1C4, C42.9(C2×C4), (C2×C6).7D14, (C2×C14).7D6, (C22×S3).D7, C14.13(C4×S3), C21⋊2(C22⋊C4), (C2×Dic7)⋊1S3, (C6×Dic7)⋊1C2, C2.4(S3×Dic7), C6.4(C2×Dic7), C3⋊1(C23.D7), C22.6(S3×D7), (C2×Dic21)⋊6C2, C6.12(C7⋊D4), C2.1(C7⋊D12), C2.2(C21⋊D4), (C2×C42).4C22, C14.12(C3⋊D4), (S3×C2×C14).1C2, SmallGroup(336,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊Dic7
G = < a,b,c,d | a6=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 284 in 68 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C7, C2×C4, C23, Dic3, C12, D6, D6, C2×C6, C14, C14, C22⋊C4, C21, C2×Dic3, C2×C12, C22×S3, Dic7, C2×C14, C2×C14, S3×C7, C42, D6⋊C4, C2×Dic7, C2×Dic7, C22×C14, C3×Dic7, Dic21, S3×C14, S3×C14, C2×C42, C23.D7, C6×Dic7, C2×Dic21, S3×C2×C14, D6⋊Dic7
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, D7, C22⋊C4, C4×S3, D12, C3⋊D4, Dic7, D14, D6⋊C4, C2×Dic7, C7⋊D4, S3×D7, C23.D7, S3×Dic7, C21⋊D4, C7⋊D12, D6⋊Dic7
►On 168 points
Generators in S168
(1 48 86 110 25 58)(2 49 87 111 26 59)(3 50 88 112 27 60)(4 51 89 99 28 61)(5 52 90 100 15 62)(6 53 91 101 16 63)(7 54 92 102 17 64)(8 55 93 103 18 65)(9 56 94 104 19 66)(10 43 95 105 20 67)(11 44 96 106 21 68)(12 45 97 107 22 69)(13 46 98 108 23 70)(14 47 85 109 24 57)(29 80 160 122 136 142)(30 81 161 123 137 143)(31 82 162 124 138 144)(32 83 163 125 139 145)(33 84 164 126 140 146)(34 71 165 113 127 147)(35 72 166 114 128 148)(36 73 167 115 129 149)(37 74 168 116 130 150)(38 75 155 117 131 151)(39 76 156 118 132 152)(40 77 157 119 133 153)(41 78 158 120 134 154)(42 79 159 121 135 141)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 43)(28 44)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(71 154)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(81 150)(82 151)(83 152)(84 153)(85 102)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 99)(97 100)(98 101)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(127 158)(128 159)(129 160)(130 161)(131 162)(132 163)(133 164)(134 165)(135 166)(136 167)(137 168)(138 155)(139 156)(140 157)
(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)
(1 157 8 164)(2 156 9 163)(3 155 10 162)(4 168 11 161)(5 167 12 160)(6 166 13 159)(7 165 14 158)(15 36 22 29)(16 35 23 42)(17 34 24 41)(18 33 25 40)(19 32 26 39)(20 31 27 38)(21 30 28 37)(43 124 50 117)(44 123 51 116)(45 122 52 115)(46 121 53 114)(47 120 54 113)(48 119 55 126)(49 118 56 125)(57 78 64 71)(58 77 65 84)(59 76 66 83)(60 75 67 82)(61 74 68 81)(62 73 69 80)(63 72 70 79)(85 134 92 127)(86 133 93 140)(87 132 94 139)(88 131 95 138)(89 130 96 137)(90 129 97 136)(91 128 98 135)(99 150 106 143)(100 149 107 142)(101 148 108 141)(102 147 109 154)(103 146 110 153)(104 145 111 152)(105 144 112 151)
G:=sub<Sym(168)| (1,48,86,110,25,58)(2,49,87,111,26,59)(3,50,88,112,27,60)(4,51,89,99,28,61)(5,52,90,100,15,62)(6,53,91,101,16,63)(7,54,92,102,17,64)(8,55,93,103,18,65)(9,56,94,104,19,66)(10,43,95,105,20,67)(11,44,96,106,21,68)(12,45,97,107,22,69)(13,46,98,108,23,70)(14,47,85,109,24,57)(29,80,160,122,136,142)(30,81,161,123,137,143)(31,82,162,124,138,144)(32,83,163,125,139,145)(33,84,164,126,140,146)(34,71,165,113,127,147)(35,72,166,114,128,148)(36,73,167,115,129,149)(37,74,168,116,130,150)(38,75,155,117,131,151)(39,76,156,118,132,152)(40,77,157,119,133,153)(41,78,158,120,134,154)(42,79,159,121,135,141), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,43)(28,44)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(71,154)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,99)(97,100)(98,101)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,155)(139,156)(140,157), (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), (1,157,8,164)(2,156,9,163)(3,155,10,162)(4,168,11,161)(5,167,12,160)(6,166,13,159)(7,165,14,158)(15,36,22,29)(16,35,23,42)(17,34,24,41)(18,33,25,40)(19,32,26,39)(20,31,27,38)(21,30,28,37)(43,124,50,117)(44,123,51,116)(45,122,52,115)(46,121,53,114)(47,120,54,113)(48,119,55,126)(49,118,56,125)(57,78,64,71)(58,77,65,84)(59,76,66,83)(60,75,67,82)(61,74,68,81)(62,73,69,80)(63,72,70,79)(85,134,92,127)(86,133,93,140)(87,132,94,139)(88,131,95,138)(89,130,96,137)(90,129,97,136)(91,128,98,135)(99,150,106,143)(100,149,107,142)(101,148,108,141)(102,147,109,154)(103,146,110,153)(104,145,111,152)(105,144,112,151)>;
G:=Group( (1,48,86,110,25,58)(2,49,87,111,26,59)(3,50,88,112,27,60)(4,51,89,99,28,61)(5,52,90,100,15,62)(6,53,91,101,16,63)(7,54,92,102,17,64)(8,55,93,103,18,65)(9,56,94,104,19,66)(10,43,95,105,20,67)(11,44,96,106,21,68)(12,45,97,107,22,69)(13,46,98,108,23,70)(14,47,85,109,24,57)(29,80,160,122,136,142)(30,81,161,123,137,143)(31,82,162,124,138,144)(32,83,163,125,139,145)(33,84,164,126,140,146)(34,71,165,113,127,147)(35,72,166,114,128,148)(36,73,167,115,129,149)(37,74,168,116,130,150)(38,75,155,117,131,151)(39,76,156,118,132,152)(40,77,157,119,133,153)(41,78,158,120,134,154)(42,79,159,121,135,141), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,43)(28,44)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(71,154)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,99)(97,100)(98,101)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,155)(139,156)(140,157), (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), (1,157,8,164)(2,156,9,163)(3,155,10,162)(4,168,11,161)(5,167,12,160)(6,166,13,159)(7,165,14,158)(15,36,22,29)(16,35,23,42)(17,34,24,41)(18,33,25,40)(19,32,26,39)(20,31,27,38)(21,30,28,37)(43,124,50,117)(44,123,51,116)(45,122,52,115)(46,121,53,114)(47,120,54,113)(48,119,55,126)(49,118,56,125)(57,78,64,71)(58,77,65,84)(59,76,66,83)(60,75,67,82)(61,74,68,81)(62,73,69,80)(63,72,70,79)(85,134,92,127)(86,133,93,140)(87,132,94,139)(88,131,95,138)(89,130,96,137)(90,129,97,136)(91,128,98,135)(99,150,106,143)(100,149,107,142)(101,148,108,141)(102,147,109,154)(103,146,110,153)(104,145,111,152)(105,144,112,151) );
G=PermutationGroup([[(1,48,86,110,25,58),(2,49,87,111,26,59),(3,50,88,112,27,60),(4,51,89,99,28,61),(5,52,90,100,15,62),(6,53,91,101,16,63),(7,54,92,102,17,64),(8,55,93,103,18,65),(9,56,94,104,19,66),(10,43,95,105,20,67),(11,44,96,106,21,68),(12,45,97,107,22,69),(13,46,98,108,23,70),(14,47,85,109,24,57),(29,80,160,122,136,142),(30,81,161,123,137,143),(31,82,162,124,138,144),(32,83,163,125,139,145),(33,84,164,126,140,146),(34,71,165,113,127,147),(35,72,166,114,128,148),(36,73,167,115,129,149),(37,74,168,116,130,150),(38,75,155,117,131,151),(39,76,156,118,132,152),(40,77,157,119,133,153),(41,78,158,120,134,154),(42,79,159,121,135,141)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,57),(8,58),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,43),(28,44),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(71,154),(72,141),(73,142),(74,143),(75,144),(76,145),(77,146),(78,147),(79,148),(80,149),(81,150),(82,151),(83,152),(84,153),(85,102),(86,103),(87,104),(88,105),(89,106),(90,107),(91,108),(92,109),(93,110),(94,111),(95,112),(96,99),(97,100),(98,101),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(127,158),(128,159),(129,160),(130,161),(131,162),(132,163),(133,164),(134,165),(135,166),(136,167),(137,168),(138,155),(139,156),(140,157)], [(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)], [(1,157,8,164),(2,156,9,163),(3,155,10,162),(4,168,11,161),(5,167,12,160),(6,166,13,159),(7,165,14,158),(15,36,22,29),(16,35,23,42),(17,34,24,41),(18,33,25,40),(19,32,26,39),(20,31,27,38),(21,30,28,37),(43,124,50,117),(44,123,51,116),(45,122,52,115),(46,121,53,114),(47,120,54,113),(48,119,55,126),(49,118,56,125),(57,78,64,71),(58,77,65,84),(59,76,66,83),(60,75,67,82),(61,74,68,81),(62,73,69,80),(63,72,70,79),(85,134,92,127),(86,133,93,140),(87,132,94,139),(88,131,95,138),(89,130,96,137),(90,129,97,136),(91,128,98,135),(99,150,106,143),(100,149,107,142),(101,148,108,141),(102,147,109,154),(103,146,110,153),(104,145,111,152),(105,144,112,151)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | ··· | 14I | 14J | ··· | 14U | 21A | 21B | 21C | 42A | ··· | 42I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 14 | 14 | 42 | 42 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D7 | C4×S3 | D12 | C3⋊D4 | Dic7 | D14 | C7⋊D4 | S3×D7 | S3×Dic7 | C21⋊D4 | C7⋊D12 |
| kernel | D6⋊Dic7 | C6×Dic7 | C2×Dic21 | S3×C2×C14 | S3×C14 | C2×Dic7 | C42 | C2×C14 | C22×S3 | C14 | C14 | C14 | D6 | C2×C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 6 | 3 | 12 | 3 | 3 | 3 | 3 |
Matrix representation of D6⋊Dic7 ►in GL4(𝔽337) generated by
| 336 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 2 | 325 |
| 0 | 0 | 253 | 336 |
| 336 | 0 | 0 | 0 |
| 266 | 1 | 0 | 0 |
| 0 | 0 | 335 | 12 |
| 0 | 0 | 84 | 2 |
| 52 | 0 | 0 | 0 |
| 183 | 175 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
| 71 | 335 | 0 | 0 |
| 161 | 266 | 0 | 0 |
| 0 | 0 | 292 | 23 |
| 0 | 0 | 161 | 45 |
G:=sub<GL(4,GF(337))| [336,0,0,0,0,336,0,0,0,0,2,253,0,0,325,336],[336,266,0,0,0,1,0,0,0,0,335,84,0,0,12,2],[52,183,0,0,0,175,0,0,0,0,336,0,0,0,0,336],[71,161,0,0,335,266,0,0,0,0,292,161,0,0,23,45] >;
D6⋊Dic7 in GAP, Magma, Sage, TeX
D_6\rtimes {\rm Dic}_7 % in TeX
G:=Group("D6:Dic7"); // GroupNames label
G:=SmallGroup(336,43);
// by ID
G=gap.SmallGroup(336,43);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,121,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^14=1,d^2=c^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations