metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2D21, C4.3D42, D84.2C2, C28.11D6, C42.36D4, C21⋊13SD16, C12.11D14, C84.3C22, C3⋊3(Q8⋊D7), (C3×Q8)⋊1D7, (C7×Q8)⋊3S3, C21⋊C8⋊3C2, (Q8×C21)⋊1C2, C7⋊3(Q8⋊2S3), C6.18(C7⋊D4), C2.6(C21⋊7D4), C14.18(C3⋊D4), SmallGroup(336,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2D21
G = < a,b,c,d | a4=c21=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Smallest permutation representation of Q8⋊2D21
►On 168 points
Generators in S168
(1 75 40 63)(2 76 41 43)(3 77 42 44)(4 78 22 45)(5 79 23 46)(6 80 24 47)(7 81 25 48)(8 82 26 49)(9 83 27 50)(10 84 28 51)(11 64 29 52)(12 65 30 53)(13 66 31 54)(14 67 32 55)(15 68 33 56)(16 69 34 57)(17 70 35 58)(18 71 36 59)(19 72 37 60)(20 73 38 61)(21 74 39 62)(85 139 111 158)(86 140 112 159)(87 141 113 160)(88 142 114 161)(89 143 115 162)(90 144 116 163)(91 145 117 164)(92 146 118 165)(93 147 119 166)(94 127 120 167)(95 128 121 168)(96 129 122 148)(97 130 123 149)(98 131 124 150)(99 132 125 151)(100 133 126 152)(101 134 106 153)(102 135 107 154)(103 136 108 155)(104 137 109 156)(105 138 110 157)
(1 117 40 91)(2 118 41 92)(3 119 42 93)(4 120 22 94)(5 121 23 95)(6 122 24 96)(7 123 25 97)(8 124 26 98)(9 125 27 99)(10 126 28 100)(11 106 29 101)(12 107 30 102)(13 108 31 103)(14 109 32 104)(15 110 33 105)(16 111 34 85)(17 112 35 86)(18 113 36 87)(19 114 37 88)(20 115 38 89)(21 116 39 90)(43 165 76 146)(44 166 77 147)(45 167 78 127)(46 168 79 128)(47 148 80 129)(48 149 81 130)(49 150 82 131)(50 151 83 132)(51 152 84 133)(52 153 64 134)(53 154 65 135)(54 155 66 136)(55 156 67 137)(56 157 68 138)(57 158 69 139)(58 159 70 140)(59 160 71 141)(60 161 72 142)(61 162 73 143)(62 163 74 144)(63 164 75 145)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(37 42)(38 41)(39 40)(43 73)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(85 129)(86 128)(87 127)(88 147)(89 146)(90 145)(91 144)(92 143)(93 142)(94 141)(95 140)(96 139)(97 138)(98 137)(99 136)(100 135)(101 134)(102 133)(103 132)(104 131)(105 130)(106 153)(107 152)(108 151)(109 150)(110 149)(111 148)(112 168)(113 167)(114 166)(115 165)(116 164)(117 163)(118 162)(119 161)(120 160)(121 159)(122 158)(123 157)(124 156)(125 155)(126 154)
G:=sub<Sym(168)| (1,75,40,63)(2,76,41,43)(3,77,42,44)(4,78,22,45)(5,79,23,46)(6,80,24,47)(7,81,25,48)(8,82,26,49)(9,83,27,50)(10,84,28,51)(11,64,29,52)(12,65,30,53)(13,66,31,54)(14,67,32,55)(15,68,33,56)(16,69,34,57)(17,70,35,58)(18,71,36,59)(19,72,37,60)(20,73,38,61)(21,74,39,62)(85,139,111,158)(86,140,112,159)(87,141,113,160)(88,142,114,161)(89,143,115,162)(90,144,116,163)(91,145,117,164)(92,146,118,165)(93,147,119,166)(94,127,120,167)(95,128,121,168)(96,129,122,148)(97,130,123,149)(98,131,124,150)(99,132,125,151)(100,133,126,152)(101,134,106,153)(102,135,107,154)(103,136,108,155)(104,137,109,156)(105,138,110,157), (1,117,40,91)(2,118,41,92)(3,119,42,93)(4,120,22,94)(5,121,23,95)(6,122,24,96)(7,123,25,97)(8,124,26,98)(9,125,27,99)(10,126,28,100)(11,106,29,101)(12,107,30,102)(13,108,31,103)(14,109,32,104)(15,110,33,105)(16,111,34,85)(17,112,35,86)(18,113,36,87)(19,114,37,88)(20,115,38,89)(21,116,39,90)(43,165,76,146)(44,166,77,147)(45,167,78,127)(46,168,79,128)(47,148,80,129)(48,149,81,130)(49,150,82,131)(50,151,83,132)(51,152,84,133)(52,153,64,134)(53,154,65,135)(54,155,66,136)(55,156,67,137)(56,157,68,138)(57,158,69,139)(58,159,70,140)(59,160,71,141)(60,161,72,142)(61,162,73,143)(62,163,74,144)(63,164,75,145), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(37,42)(38,41)(39,40)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(85,129)(86,128)(87,127)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131)(105,130)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,168)(113,167)(114,166)(115,165)(116,164)(117,163)(118,162)(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)>;
G:=Group( (1,75,40,63)(2,76,41,43)(3,77,42,44)(4,78,22,45)(5,79,23,46)(6,80,24,47)(7,81,25,48)(8,82,26,49)(9,83,27,50)(10,84,28,51)(11,64,29,52)(12,65,30,53)(13,66,31,54)(14,67,32,55)(15,68,33,56)(16,69,34,57)(17,70,35,58)(18,71,36,59)(19,72,37,60)(20,73,38,61)(21,74,39,62)(85,139,111,158)(86,140,112,159)(87,141,113,160)(88,142,114,161)(89,143,115,162)(90,144,116,163)(91,145,117,164)(92,146,118,165)(93,147,119,166)(94,127,120,167)(95,128,121,168)(96,129,122,148)(97,130,123,149)(98,131,124,150)(99,132,125,151)(100,133,126,152)(101,134,106,153)(102,135,107,154)(103,136,108,155)(104,137,109,156)(105,138,110,157), (1,117,40,91)(2,118,41,92)(3,119,42,93)(4,120,22,94)(5,121,23,95)(6,122,24,96)(7,123,25,97)(8,124,26,98)(9,125,27,99)(10,126,28,100)(11,106,29,101)(12,107,30,102)(13,108,31,103)(14,109,32,104)(15,110,33,105)(16,111,34,85)(17,112,35,86)(18,113,36,87)(19,114,37,88)(20,115,38,89)(21,116,39,90)(43,165,76,146)(44,166,77,147)(45,167,78,127)(46,168,79,128)(47,148,80,129)(48,149,81,130)(49,150,82,131)(50,151,83,132)(51,152,84,133)(52,153,64,134)(53,154,65,135)(54,155,66,136)(55,156,67,137)(56,157,68,138)(57,158,69,139)(58,159,70,140)(59,160,71,141)(60,161,72,142)(61,162,73,143)(62,163,74,144)(63,164,75,145), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(37,42)(38,41)(39,40)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(85,129)(86,128)(87,127)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131)(105,130)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,168)(113,167)(114,166)(115,165)(116,164)(117,163)(118,162)(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154) );
G=PermutationGroup([[(1,75,40,63),(2,76,41,43),(3,77,42,44),(4,78,22,45),(5,79,23,46),(6,80,24,47),(7,81,25,48),(8,82,26,49),(9,83,27,50),(10,84,28,51),(11,64,29,52),(12,65,30,53),(13,66,31,54),(14,67,32,55),(15,68,33,56),(16,69,34,57),(17,70,35,58),(18,71,36,59),(19,72,37,60),(20,73,38,61),(21,74,39,62),(85,139,111,158),(86,140,112,159),(87,141,113,160),(88,142,114,161),(89,143,115,162),(90,144,116,163),(91,145,117,164),(92,146,118,165),(93,147,119,166),(94,127,120,167),(95,128,121,168),(96,129,122,148),(97,130,123,149),(98,131,124,150),(99,132,125,151),(100,133,126,152),(101,134,106,153),(102,135,107,154),(103,136,108,155),(104,137,109,156),(105,138,110,157)], [(1,117,40,91),(2,118,41,92),(3,119,42,93),(4,120,22,94),(5,121,23,95),(6,122,24,96),(7,123,25,97),(8,124,26,98),(9,125,27,99),(10,126,28,100),(11,106,29,101),(12,107,30,102),(13,108,31,103),(14,109,32,104),(15,110,33,105),(16,111,34,85),(17,112,35,86),(18,113,36,87),(19,114,37,88),(20,115,38,89),(21,116,39,90),(43,165,76,146),(44,166,77,147),(45,167,78,127),(46,168,79,128),(47,148,80,129),(48,149,81,130),(49,150,82,131),(50,151,83,132),(51,152,84,133),(52,153,64,134),(53,154,65,135),(54,155,66,136),(55,156,67,137),(56,157,68,138),(57,158,69,139),(58,159,70,140),(59,160,71,141),(60,161,72,142),(61,162,73,143),(62,163,74,144),(63,164,75,145)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(37,42),(38,41),(39,40),(43,73),(44,72),(45,71),(46,70),(47,69),(48,68),(49,67),(50,66),(51,65),(52,64),(53,84),(54,83),(55,82),(56,81),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74),(85,129),(86,128),(87,127),(88,147),(89,146),(90,145),(91,144),(92,143),(93,142),(94,141),(95,140),(96,139),(97,138),(98,137),(99,136),(100,135),(101,134),(102,133),(103,132),(104,131),(105,130),(106,153),(107,152),(108,151),(109,150),(110,149),(111,148),(112,168),(113,167),(114,166),(115,165),(116,164),(117,163),(118,162),(119,161),(120,160),(121,159),(122,158),(123,157),(124,156),(125,155),(126,154)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 12C | 14A | 14B | 14C | 21A | ··· | 21F | 28A | ··· | 28I | 42A | ··· | 42F | 84A | ··· | 84R |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 84 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 42 | 42 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 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 | Q8⋊2S3 | Q8⋊D7 | Q8⋊2D21 |
| kernel | Q8⋊2D21 | C21⋊C8 | D84 | Q8×C21 | C7×Q8 | C42 | C28 | C3×Q8 | C21 | C14 | C12 | Q8 | 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 Q8⋊2D21 ►in GL6(𝔽337)
| 336 | 0 | 0 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 336 | 48 |
| 0 | 0 | 0 | 0 | 14 | 1 |
| 158 | 205 | 0 | 0 | 0 | 0 |
| 304 | 179 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 |
| 0 | 0 | 0 | 0 | 24 | 0 |
| 85 | 112 | 0 | 0 | 0 | 0 |
| 28 | 251 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 34 | 0 | 0 |
| 0 | 0 | 303 | 144 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 85 | 112 | 0 | 0 | 0 | 0 |
| 104 | 252 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 303 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 323 | 336 |
G:=sub<GL(6,GF(337))| [336,0,0,0,0,0,0,336,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,336,14,0,0,0,0,48,1],[158,304,0,0,0,0,205,179,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,0,24,0,0,0,0,14,0],[85,28,0,0,0,0,112,251,0,0,0,0,0,0,336,303,0,0,0,0,34,144,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[85,104,0,0,0,0,112,252,0,0,0,0,0,0,336,303,0,0,0,0,0,1,0,0,0,0,0,0,1,323,0,0,0,0,0,336] >;
Q8⋊2D21 in GAP, Magma, Sage, TeX
Q_8\rtimes_2D_{21} % in TeX
G:=Group("Q8:2D21"); // GroupNames label
G:=SmallGroup(336,103);
// by ID
G=gap.SmallGroup(336,103);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,73,55,218,116,50,964,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^21=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
Export