metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊7Q8, C42.149D10, C10.1312+ 1+4, C20⋊Q8⋊36C2, C4.16(Q8×D5), C5⋊7(D4⋊3Q8), C42.C2⋊5D5, C20.51(C2×Q8), C4⋊C4.205D10, (C4×D20).24C2, D10.23(C2×Q8), D10⋊Q8⋊34C2, D10⋊2Q8⋊35C2, (C4×Dic10)⋊47C2, (C2×C20).88C23, C4.Dic10⋊34C2, D20⋊8C4.11C2, C10.43(C22×Q8), (C2×C10).234C24, (C4×C20).194C22, C2.56(D4⋊8D10), Dic5.45(C4○D4), (C2×D20).275C22, C4⋊Dic5.379C22, C22.255(C23×D5), D10⋊C4.40C22, (C2×Dic5).122C23, (C4×Dic5).149C22, (C22×D5).231C23, (C2×Dic10).259C22, C10.D4.144C22, (D5×C4⋊C4)⋊35C2, C2.26(C2×Q8×D5), C2.85(D5×C4○D4), (C5×C42.C2)⋊7C2, C10.196(C2×C4○D4), (C2×C4×D5).268C22, (C2×C4).78(C22×D5), (C5×C4⋊C4).189C22, SmallGroup(320,1362)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊7Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a9, cbc-1=a10b, dbd-1=a18b, dcd-1=c-1 >
Subgroups: 806 in 228 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, D4⋊3Q8, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×D20, C4×Dic10, C4×D20, C20⋊Q8, C4.Dic10, D5×C4⋊C4, D20⋊8C4, D10⋊Q8, D10⋊2Q8, C5×C42.C2, D20⋊7Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, 2+ 1+4, C22×D5, D4⋊3Q8, Q8×D5, C23×D5, C2×Q8×D5, D5×C4○D4, D4⋊8D10, D20⋊7Q8
►On 160 points
Generators in S160
(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)
(1 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 97)(22 96)(23 95)(24 94)(25 93)(26 92)(27 91)(28 90)(29 89)(30 88)(31 87)(32 86)(33 85)(34 84)(35 83)(36 82)(37 81)(38 100)(39 99)(40 98)(41 128)(42 127)(43 126)(44 125)(45 124)(46 123)(47 122)(48 121)(49 140)(50 139)(51 138)(52 137)(53 136)(54 135)(55 134)(56 133)(57 132)(58 131)(59 130)(60 129)(101 156)(102 155)(103 154)(104 153)(105 152)(106 151)(107 150)(108 149)(109 148)(110 147)(111 146)(112 145)(113 144)(114 143)(115 142)(116 141)(117 160)(118 159)(119 158)(120 157)
(1 135 70 60)(2 136 71 41)(3 137 72 42)(4 138 73 43)(5 139 74 44)(6 140 75 45)(7 121 76 46)(8 122 77 47)(9 123 78 48)(10 124 79 49)(11 125 80 50)(12 126 61 51)(13 127 62 52)(14 128 63 53)(15 129 64 54)(16 130 65 55)(17 131 66 56)(18 132 67 57)(19 133 68 58)(20 134 69 59)(21 101 93 142)(22 102 94 143)(23 103 95 144)(24 104 96 145)(25 105 97 146)(26 106 98 147)(27 107 99 148)(28 108 100 149)(29 109 81 150)(30 110 82 151)(31 111 83 152)(32 112 84 153)(33 113 85 154)(34 114 86 155)(35 115 87 156)(36 116 88 157)(37 117 89 158)(38 118 90 159)(39 119 91 160)(40 120 92 141)
(1 88 70 36)(2 97 71 25)(3 86 72 34)(4 95 73 23)(5 84 74 32)(6 93 75 21)(7 82 76 30)(8 91 77 39)(9 100 78 28)(10 89 79 37)(11 98 80 26)(12 87 61 35)(13 96 62 24)(14 85 63 33)(15 94 64 22)(16 83 65 31)(17 92 66 40)(18 81 67 29)(19 90 68 38)(20 99 69 27)(41 146 136 105)(42 155 137 114)(43 144 138 103)(44 153 139 112)(45 142 140 101)(46 151 121 110)(47 160 122 119)(48 149 123 108)(49 158 124 117)(50 147 125 106)(51 156 126 115)(52 145 127 104)(53 154 128 113)(54 143 129 102)(55 152 130 111)(56 141 131 120)(57 150 132 109)(58 159 133 118)(59 148 134 107)(60 157 135 116)
G:=sub<Sym(160)| (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), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,97)(22,96)(23,95)(24,94)(25,93)(26,92)(27,91)(28,90)(29,89)(30,88)(31,87)(32,86)(33,85)(34,84)(35,83)(36,82)(37,81)(38,100)(39,99)(40,98)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,140)(50,139)(51,138)(52,137)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(101,156)(102,155)(103,154)(104,153)(105,152)(106,151)(107,150)(108,149)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,160)(118,159)(119,158)(120,157), (1,135,70,60)(2,136,71,41)(3,137,72,42)(4,138,73,43)(5,139,74,44)(6,140,75,45)(7,121,76,46)(8,122,77,47)(9,123,78,48)(10,124,79,49)(11,125,80,50)(12,126,61,51)(13,127,62,52)(14,128,63,53)(15,129,64,54)(16,130,65,55)(17,131,66,56)(18,132,67,57)(19,133,68,58)(20,134,69,59)(21,101,93,142)(22,102,94,143)(23,103,95,144)(24,104,96,145)(25,105,97,146)(26,106,98,147)(27,107,99,148)(28,108,100,149)(29,109,81,150)(30,110,82,151)(31,111,83,152)(32,112,84,153)(33,113,85,154)(34,114,86,155)(35,115,87,156)(36,116,88,157)(37,117,89,158)(38,118,90,159)(39,119,91,160)(40,120,92,141), (1,88,70,36)(2,97,71,25)(3,86,72,34)(4,95,73,23)(5,84,74,32)(6,93,75,21)(7,82,76,30)(8,91,77,39)(9,100,78,28)(10,89,79,37)(11,98,80,26)(12,87,61,35)(13,96,62,24)(14,85,63,33)(15,94,64,22)(16,83,65,31)(17,92,66,40)(18,81,67,29)(19,90,68,38)(20,99,69,27)(41,146,136,105)(42,155,137,114)(43,144,138,103)(44,153,139,112)(45,142,140,101)(46,151,121,110)(47,160,122,119)(48,149,123,108)(49,158,124,117)(50,147,125,106)(51,156,126,115)(52,145,127,104)(53,154,128,113)(54,143,129,102)(55,152,130,111)(56,141,131,120)(57,150,132,109)(58,159,133,118)(59,148,134,107)(60,157,135,116)>;
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), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,97)(22,96)(23,95)(24,94)(25,93)(26,92)(27,91)(28,90)(29,89)(30,88)(31,87)(32,86)(33,85)(34,84)(35,83)(36,82)(37,81)(38,100)(39,99)(40,98)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,140)(50,139)(51,138)(52,137)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(101,156)(102,155)(103,154)(104,153)(105,152)(106,151)(107,150)(108,149)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,160)(118,159)(119,158)(120,157), (1,135,70,60)(2,136,71,41)(3,137,72,42)(4,138,73,43)(5,139,74,44)(6,140,75,45)(7,121,76,46)(8,122,77,47)(9,123,78,48)(10,124,79,49)(11,125,80,50)(12,126,61,51)(13,127,62,52)(14,128,63,53)(15,129,64,54)(16,130,65,55)(17,131,66,56)(18,132,67,57)(19,133,68,58)(20,134,69,59)(21,101,93,142)(22,102,94,143)(23,103,95,144)(24,104,96,145)(25,105,97,146)(26,106,98,147)(27,107,99,148)(28,108,100,149)(29,109,81,150)(30,110,82,151)(31,111,83,152)(32,112,84,153)(33,113,85,154)(34,114,86,155)(35,115,87,156)(36,116,88,157)(37,117,89,158)(38,118,90,159)(39,119,91,160)(40,120,92,141), (1,88,70,36)(2,97,71,25)(3,86,72,34)(4,95,73,23)(5,84,74,32)(6,93,75,21)(7,82,76,30)(8,91,77,39)(9,100,78,28)(10,89,79,37)(11,98,80,26)(12,87,61,35)(13,96,62,24)(14,85,63,33)(15,94,64,22)(16,83,65,31)(17,92,66,40)(18,81,67,29)(19,90,68,38)(20,99,69,27)(41,146,136,105)(42,155,137,114)(43,144,138,103)(44,153,139,112)(45,142,140,101)(46,151,121,110)(47,160,122,119)(48,149,123,108)(49,158,124,117)(50,147,125,106)(51,156,126,115)(52,145,127,104)(53,154,128,113)(54,143,129,102)(55,152,130,111)(56,141,131,120)(57,150,132,109)(58,159,133,118)(59,148,134,107)(60,157,135,116) );
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)], [(1,74),(2,73),(3,72),(4,71),(5,70),(6,69),(7,68),(8,67),(9,66),(10,65),(11,64),(12,63),(13,62),(14,61),(15,80),(16,79),(17,78),(18,77),(19,76),(20,75),(21,97),(22,96),(23,95),(24,94),(25,93),(26,92),(27,91),(28,90),(29,89),(30,88),(31,87),(32,86),(33,85),(34,84),(35,83),(36,82),(37,81),(38,100),(39,99),(40,98),(41,128),(42,127),(43,126),(44,125),(45,124),(46,123),(47,122),(48,121),(49,140),(50,139),(51,138),(52,137),(53,136),(54,135),(55,134),(56,133),(57,132),(58,131),(59,130),(60,129),(101,156),(102,155),(103,154),(104,153),(105,152),(106,151),(107,150),(108,149),(109,148),(110,147),(111,146),(112,145),(113,144),(114,143),(115,142),(116,141),(117,160),(118,159),(119,158),(120,157)], [(1,135,70,60),(2,136,71,41),(3,137,72,42),(4,138,73,43),(5,139,74,44),(6,140,75,45),(7,121,76,46),(8,122,77,47),(9,123,78,48),(10,124,79,49),(11,125,80,50),(12,126,61,51),(13,127,62,52),(14,128,63,53),(15,129,64,54),(16,130,65,55),(17,131,66,56),(18,132,67,57),(19,133,68,58),(20,134,69,59),(21,101,93,142),(22,102,94,143),(23,103,95,144),(24,104,96,145),(25,105,97,146),(26,106,98,147),(27,107,99,148),(28,108,100,149),(29,109,81,150),(30,110,82,151),(31,111,83,152),(32,112,84,153),(33,113,85,154),(34,114,86,155),(35,115,87,156),(36,116,88,157),(37,117,89,158),(38,118,90,159),(39,119,91,160),(40,120,92,141)], [(1,88,70,36),(2,97,71,25),(3,86,72,34),(4,95,73,23),(5,84,74,32),(6,93,75,21),(7,82,76,30),(8,91,77,39),(9,100,78,28),(10,89,79,37),(11,98,80,26),(12,87,61,35),(13,96,62,24),(14,85,63,33),(15,94,64,22),(16,83,65,31),(17,92,66,40),(18,81,67,29),(19,90,68,38),(20,99,69,27),(41,146,136,105),(42,155,137,114),(43,144,138,103),(44,153,139,112),(45,142,140,101),(46,151,121,110),(47,160,122,119),(48,149,123,108),(49,158,124,117),(50,147,125,106),(51,156,126,115),(52,145,127,104),(53,154,128,113),(54,143,129,102),(55,152,130,111),(56,141,131,120),(57,150,132,109),(58,159,133,118),(59,148,134,107),(60,157,135,116)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | 2+ 1+4 | Q8×D5 | D5×C4○D4 | D4⋊8D10 |
| kernel | D20⋊7Q8 | C4×Dic10 | C4×D20 | C20⋊Q8 | C4.Dic10 | D5×C4⋊C4 | D20⋊8C4 | D10⋊Q8 | D10⋊2Q8 | C5×C42.C2 | D20 | C42.C2 | Dic5 | C42 | C4⋊C4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 4 | 2 | 4 | 2 | 12 | 1 | 4 | 4 | 4 |
Matrix representation of D20⋊7Q8 ►in GL6(𝔽41)
| 34 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 34 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 13 |
| 0 | 0 | 0 | 0 | 34 | 34 |
G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,7,34,0,0,0,0,13,34] >;
D20⋊7Q8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_7Q_8 % in TeX
G:=Group("D20:7Q8"); // GroupNames label
G:=SmallGroup(320,1362);
// by ID
G=gap.SmallGroup(320,1362);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,219,184,1571,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^9,c*b*c^-1=a^10*b,d*b*d^-1=a^18*b,d*c*d^-1=c^-1>;
// generators/relations