metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.4D14, Dic7.9D6, C42.19C23, Dic3.6D14, D42.4C22, Dic21.6C22, C21⋊Q8⋊6C2, C3⋊D4⋊3D7, D21⋊C4⋊4C2, C7⋊4(C4○D12), C21⋊7D4⋊4C2, C7⋊D12⋊4C2, (C2×C14).2D6, C21⋊10(C4○D4), C3⋊3(D4⋊2D7), (C6×Dic7)⋊4C2, (S3×Dic7)⋊4C2, (C2×Dic7)⋊3S3, (C2×C6).14D14, C22.2(S3×D7), C6.19(C22×D7), (S3×C14).4C22, C14.19(C22×S3), (C2×C42).13C22, (C7×Dic3).6C22, (C3×Dic7).11C22, C2.20(C2×S3×D7), (C7×C3⋊D4)⋊2C2, SmallGroup(336,155)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3.D14
G = < a,b,c,d | a42=c2=d2=1, b2=a21, bab-1=a13, cac=a29, ad=da, bc=cb, bd=db, dcd=a21c >
Subgroups: 428 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, Q8, Dic3, Dic3, C12, D6, D6, C2×C6, D7, C14, C14, C4○D4, C21, Dic6, C4×S3, D12, C3⋊D4, C3⋊D4, C2×C12, Dic7, Dic7, C28, D14, C2×C14, C2×C14, S3×C7, D21, C42, C42, C4○D12, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C7×D4, C7×Dic3, C3×Dic7, Dic21, S3×C14, D42, C2×C42, D4⋊2D7, S3×Dic7, D21⋊C4, C7⋊D12, C21⋊Q8, C6×Dic7, C7×C3⋊D4, C21⋊7D4, Dic3.D14
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, C4○D12, C22×D7, S3×D7, D4⋊2D7, C2×S3×D7, Dic3.D14
►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)
(1 134 22 155)(2 147 23 168)(3 160 24 139)(4 131 25 152)(5 144 26 165)(6 157 27 136)(7 128 28 149)(8 141 29 162)(9 154 30 133)(10 167 31 146)(11 138 32 159)(12 151 33 130)(13 164 34 143)(14 135 35 156)(15 148 36 127)(16 161 37 140)(17 132 38 153)(18 145 39 166)(19 158 40 137)(20 129 41 150)(21 142 42 163)(43 99 64 120)(44 112 65 91)(45 125 66 104)(46 96 67 117)(47 109 68 88)(48 122 69 101)(49 93 70 114)(50 106 71 85)(51 119 72 98)(52 90 73 111)(53 103 74 124)(54 116 75 95)(55 87 76 108)(56 100 77 121)(57 113 78 92)(58 126 79 105)(59 97 80 118)(60 110 81 89)(61 123 82 102)(62 94 83 115)(63 107 84 86)
(1 99)(2 86)(3 115)(4 102)(5 89)(6 118)(7 105)(8 92)(9 121)(10 108)(11 95)(12 124)(13 111)(14 98)(15 85)(16 114)(17 101)(18 88)(19 117)(20 104)(21 91)(22 120)(23 107)(24 94)(25 123)(26 110)(27 97)(28 126)(29 113)(30 100)(31 87)(32 116)(33 103)(34 90)(35 119)(36 106)(37 93)(38 122)(39 109)(40 96)(41 125)(42 112)(43 155)(44 142)(45 129)(46 158)(47 145)(48 132)(49 161)(50 148)(51 135)(52 164)(53 151)(54 138)(55 167)(56 154)(57 141)(58 128)(59 157)(60 144)(61 131)(62 160)(63 147)(64 134)(65 163)(66 150)(67 137)(68 166)(69 153)(70 140)(71 127)(72 156)(73 143)(74 130)(75 159)(76 146)(77 133)(78 162)(79 149)(80 136)(81 165)(82 152)(83 139)(84 168)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)(101 157)(102 158)(103 159)(104 160)(105 161)(106 162)(107 163)(108 164)(109 165)(110 166)(111 167)(112 168)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)
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), (1,134,22,155)(2,147,23,168)(3,160,24,139)(4,131,25,152)(5,144,26,165)(6,157,27,136)(7,128,28,149)(8,141,29,162)(9,154,30,133)(10,167,31,146)(11,138,32,159)(12,151,33,130)(13,164,34,143)(14,135,35,156)(15,148,36,127)(16,161,37,140)(17,132,38,153)(18,145,39,166)(19,158,40,137)(20,129,41,150)(21,142,42,163)(43,99,64,120)(44,112,65,91)(45,125,66,104)(46,96,67,117)(47,109,68,88)(48,122,69,101)(49,93,70,114)(50,106,71,85)(51,119,72,98)(52,90,73,111)(53,103,74,124)(54,116,75,95)(55,87,76,108)(56,100,77,121)(57,113,78,92)(58,126,79,105)(59,97,80,118)(60,110,81,89)(61,123,82,102)(62,94,83,115)(63,107,84,86), (1,99)(2,86)(3,115)(4,102)(5,89)(6,118)(7,105)(8,92)(9,121)(10,108)(11,95)(12,124)(13,111)(14,98)(15,85)(16,114)(17,101)(18,88)(19,117)(20,104)(21,91)(22,120)(23,107)(24,94)(25,123)(26,110)(27,97)(28,126)(29,113)(30,100)(31,87)(32,116)(33,103)(34,90)(35,119)(36,106)(37,93)(38,122)(39,109)(40,96)(41,125)(42,112)(43,155)(44,142)(45,129)(46,158)(47,145)(48,132)(49,161)(50,148)(51,135)(52,164)(53,151)(54,138)(55,167)(56,154)(57,141)(58,128)(59,157)(60,144)(61,131)(62,160)(63,147)(64,134)(65,163)(66,150)(67,137)(68,166)(69,153)(70,140)(71,127)(72,156)(73,143)(74,130)(75,159)(76,146)(77,133)(78,162)(79,149)(80,136)(81,165)(82,152)(83,139)(84,168), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156)(101,157)(102,158)(103,159)(104,160)(105,161)(106,162)(107,163)(108,164)(109,165)(110,166)(111,167)(112,168)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)>;
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), (1,134,22,155)(2,147,23,168)(3,160,24,139)(4,131,25,152)(5,144,26,165)(6,157,27,136)(7,128,28,149)(8,141,29,162)(9,154,30,133)(10,167,31,146)(11,138,32,159)(12,151,33,130)(13,164,34,143)(14,135,35,156)(15,148,36,127)(16,161,37,140)(17,132,38,153)(18,145,39,166)(19,158,40,137)(20,129,41,150)(21,142,42,163)(43,99,64,120)(44,112,65,91)(45,125,66,104)(46,96,67,117)(47,109,68,88)(48,122,69,101)(49,93,70,114)(50,106,71,85)(51,119,72,98)(52,90,73,111)(53,103,74,124)(54,116,75,95)(55,87,76,108)(56,100,77,121)(57,113,78,92)(58,126,79,105)(59,97,80,118)(60,110,81,89)(61,123,82,102)(62,94,83,115)(63,107,84,86), (1,99)(2,86)(3,115)(4,102)(5,89)(6,118)(7,105)(8,92)(9,121)(10,108)(11,95)(12,124)(13,111)(14,98)(15,85)(16,114)(17,101)(18,88)(19,117)(20,104)(21,91)(22,120)(23,107)(24,94)(25,123)(26,110)(27,97)(28,126)(29,113)(30,100)(31,87)(32,116)(33,103)(34,90)(35,119)(36,106)(37,93)(38,122)(39,109)(40,96)(41,125)(42,112)(43,155)(44,142)(45,129)(46,158)(47,145)(48,132)(49,161)(50,148)(51,135)(52,164)(53,151)(54,138)(55,167)(56,154)(57,141)(58,128)(59,157)(60,144)(61,131)(62,160)(63,147)(64,134)(65,163)(66,150)(67,137)(68,166)(69,153)(70,140)(71,127)(72,156)(73,143)(74,130)(75,159)(76,146)(77,133)(78,162)(79,149)(80,136)(81,165)(82,152)(83,139)(84,168), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156)(101,157)(102,158)(103,159)(104,160)(105,161)(106,162)(107,163)(108,164)(109,165)(110,166)(111,167)(112,168)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140) );
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)], [(1,134,22,155),(2,147,23,168),(3,160,24,139),(4,131,25,152),(5,144,26,165),(6,157,27,136),(7,128,28,149),(8,141,29,162),(9,154,30,133),(10,167,31,146),(11,138,32,159),(12,151,33,130),(13,164,34,143),(14,135,35,156),(15,148,36,127),(16,161,37,140),(17,132,38,153),(18,145,39,166),(19,158,40,137),(20,129,41,150),(21,142,42,163),(43,99,64,120),(44,112,65,91),(45,125,66,104),(46,96,67,117),(47,109,68,88),(48,122,69,101),(49,93,70,114),(50,106,71,85),(51,119,72,98),(52,90,73,111),(53,103,74,124),(54,116,75,95),(55,87,76,108),(56,100,77,121),(57,113,78,92),(58,126,79,105),(59,97,80,118),(60,110,81,89),(61,123,82,102),(62,94,83,115),(63,107,84,86)], [(1,99),(2,86),(3,115),(4,102),(5,89),(6,118),(7,105),(8,92),(9,121),(10,108),(11,95),(12,124),(13,111),(14,98),(15,85),(16,114),(17,101),(18,88),(19,117),(20,104),(21,91),(22,120),(23,107),(24,94),(25,123),(26,110),(27,97),(28,126),(29,113),(30,100),(31,87),(32,116),(33,103),(34,90),(35,119),(36,106),(37,93),(38,122),(39,109),(40,96),(41,125),(42,112),(43,155),(44,142),(45,129),(46,158),(47,145),(48,132),(49,161),(50,148),(51,135),(52,164),(53,151),(54,138),(55,167),(56,154),(57,141),(58,128),(59,157),(60,144),(61,131),(62,160),(63,147),(64,134),(65,163),(66,150),(67,137),(68,166),(69,153),(70,140),(71,127),(72,156),(73,143),(74,130),(75,159),(76,146),(77,133),(78,162),(79,149),(80,136),(81,165),(82,152),(83,139),(84,168)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(85,141),(86,142),(87,143),(88,144),(89,145),(90,146),(91,147),(92,148),(93,149),(94,150),(95,151),(96,152),(97,153),(98,154),(99,155),(100,156),(101,157),(102,158),(103,159),(104,160),(105,161),(106,162),(107,163),(108,164),(109,165),(110,166),(111,167),(112,168),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 21A | 21B | 21C | 28A | 28B | 28C | 42A | ··· | 42I |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 21 | 21 | 21 | 28 | 28 | 28 | 42 | ··· | 42 |
| size | 1 | 1 | 2 | 6 | 42 | 2 | 6 | 7 | 7 | 14 | 42 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 4 | 4 | 4 | 12 | 12 | 12 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D7 | C4○D4 | D14 | D14 | D14 | C4○D12 | S3×D7 | D4⋊2D7 | C2×S3×D7 | Dic3.D14 |
| kernel | Dic3.D14 | S3×Dic7 | D21⋊C4 | C7⋊D12 | C21⋊Q8 | C6×Dic7 | C7×C3⋊D4 | C21⋊7D4 | C2×Dic7 | Dic7 | C2×C14 | C3⋊D4 | C21 | Dic3 | D6 | C2×C6 | C7 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 3 | 3 | 3 | 4 | 3 | 3 | 3 | 6 |
Matrix representation of Dic3.D14 ►in GL4(𝔽337) generated by
| 0 | 1 | 0 | 0 |
| 336 | 1 | 0 | 0 |
| 0 | 0 | 228 | 336 |
| 0 | 0 | 229 | 336 |
| 189 | 0 | 0 | 0 |
| 0 | 189 | 0 | 0 |
| 0 | 0 | 228 | 336 |
| 0 | 0 | 85 | 109 |
| 322 | 322 | 0 | 0 |
| 307 | 15 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 198 | 278 | 0 | 0 |
| 59 | 139 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
G:=sub<GL(4,GF(337))| [0,336,0,0,1,1,0,0,0,0,228,229,0,0,336,336],[189,0,0,0,0,189,0,0,0,0,228,85,0,0,336,109],[322,307,0,0,322,15,0,0,0,0,1,0,0,0,0,1],[198,59,0,0,278,139,0,0,0,0,336,0,0,0,0,336] >;
Dic3.D14 in GAP, Magma, Sage, TeX
{\rm Dic}_3.D_{14} % in TeX
G:=Group("Dic3.D14"); // GroupNames label
G:=SmallGroup(336,155);
// by ID
G=gap.SmallGroup(336,155);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,116,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^42=c^2=d^2=1,b^2=a^21,b*a*b^-1=a^13,c*a*c=a^29,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^21*c>;
// generators/relations