metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20:22D4, C10.182- 1+4, C22:Q8:8D5, C5:5(D4:6D4), C4.112(D4xD5), C4:C4.189D10, D10.44(C2xD4), C20.235(C2xD4), D10:D4:25C2, D20:8C4:26C2, D10:2Q8:25C2, Dic5:5(C4oD4), (C2xQ8).126D10, C22:C4.16D10, C10.77(C22xD4), Dic5:Q8:14C2, (C2xC20).503C23, (C2xC10).175C24, (C22xC4).237D10, D10.12D4:25C2, D10.13D4:18C2, (C2xD20).155C22, C4:Dic5.215C22, (Q8xC10).107C22, C23.119(C22xD5), C22.196(C23xD5), D10:C4.23C22, (C22xC10).203C23, (C22xC20).255C22, (C4xDic5).113C22, (C2xDic5).244C23, C10.D4.27C22, (C22xD5).207C23, C23.D5.116C22, C2.19(Q8.10D10), (C2xDic10).302C22, C2.50(C2xD4xD5), (D5xC4:C4):26C2, (C4xC5:D4):23C2, C2.49(D5xC4oD4), (C2xC4oD20):24C2, (C2xQ8:2D5):8C2, (C5xC22:Q8):11C2, C10.161(C2xC4oD4), (C2xC4xD5).104C22, (C2xC4).48(C22xD5), (C5xC4:C4).158C22, (C2xC5:D4).131C22, (C5xC22:C4).30C22, SmallGroup(320,1303)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20:22D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a9, cbc-1=a8b, dbd=a18b, dcd=c-1 >
Subgroups: 1078 in 292 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, D5, C10, C10, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C2xQ8, C4oD4, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xC4:C4, C4xD4, C4:D4, C22:Q8, C22:Q8, C22.D4, C4:Q8, C2xC4oD4, Dic10, C4xD5, D20, D20, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C2xC20, C5xQ8, C22xD5, C22xD5, C22xC10, D4:6D4, C4xDic5, C10.D4, C10.D4, C4:Dic5, D10:C4, D10:C4, C23.D5, C5xC22:C4, C5xC4:C4, C5xC4:C4, C2xDic10, C2xC4xD5, C2xC4xD5, C2xD20, C2xD20, C4oD20, Q8:2D5, C2xC5:D4, C2xC5:D4, C22xC20, Q8xC10, D10.12D4, D10:D4, D5xC4:C4, D20:8C4, D10.13D4, D10:2Q8, C4xC5:D4, Dic5:Q8, C5xC22:Q8, C2xC4oD20, C2xQ8:2D5, D20:22D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, C24, D10, C22xD4, C2xC4oD4, 2- 1+4, C22xD5, D4:6D4, D4xD5, C23xD5, C2xD4xD5, Q8.10D10, D5xC4oD4, D20:22D4
►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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 40)(19 39)(20 38)(41 85)(42 84)(43 83)(44 82)(45 81)(46 100)(47 99)(48 98)(49 97)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)(61 116)(62 115)(63 114)(64 113)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(73 104)(74 103)(75 102)(76 101)(77 120)(78 119)(79 118)(80 117)(121 148)(122 147)(123 146)(124 145)(125 144)(126 143)(127 142)(128 141)(129 160)(130 159)(131 158)(132 157)(133 156)(134 155)(135 154)(136 153)(137 152)(138 151)(139 150)(140 149)
(1 152 59 67)(2 141 60 76)(3 150 41 65)(4 159 42 74)(5 148 43 63)(6 157 44 72)(7 146 45 61)(8 155 46 70)(9 144 47 79)(10 153 48 68)(11 142 49 77)(12 151 50 66)(13 160 51 75)(14 149 52 64)(15 158 53 73)(16 147 54 62)(17 156 55 71)(18 145 56 80)(19 154 57 69)(20 143 58 78)(21 125 91 118)(22 134 92 107)(23 123 93 116)(24 132 94 105)(25 121 95 114)(26 130 96 103)(27 139 97 112)(28 128 98 101)(29 137 99 110)(30 126 100 119)(31 135 81 108)(32 124 82 117)(33 133 83 106)(34 122 84 115)(35 131 85 104)(36 140 86 113)(37 129 87 102)(38 138 88 111)(39 127 89 120)(40 136 90 109)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 35)(22 24)(23 33)(25 31)(26 40)(27 29)(28 38)(30 36)(32 34)(37 39)(41 57)(42 46)(43 55)(45 53)(47 51)(48 60)(50 58)(52 56)(61 158)(62 147)(63 156)(64 145)(65 154)(66 143)(67 152)(68 141)(69 150)(70 159)(71 148)(72 157)(73 146)(74 155)(75 144)(76 153)(77 142)(78 151)(79 160)(80 149)(81 95)(82 84)(83 93)(85 91)(86 100)(87 89)(88 98)(90 96)(92 94)(97 99)(101 138)(102 127)(103 136)(104 125)(105 134)(106 123)(107 132)(108 121)(109 130)(110 139)(111 128)(112 137)(113 126)(114 135)(115 124)(116 133)(117 122)(118 131)(119 140)(120 129)
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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,40)(19,39)(20,38)(41,85)(42,84)(43,83)(44,82)(45,81)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,120)(78,119)(79,118)(80,117)(121,148)(122,147)(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149), (1,152,59,67)(2,141,60,76)(3,150,41,65)(4,159,42,74)(5,148,43,63)(6,157,44,72)(7,146,45,61)(8,155,46,70)(9,144,47,79)(10,153,48,68)(11,142,49,77)(12,151,50,66)(13,160,51,75)(14,149,52,64)(15,158,53,73)(16,147,54,62)(17,156,55,71)(18,145,56,80)(19,154,57,69)(20,143,58,78)(21,125,91,118)(22,134,92,107)(23,123,93,116)(24,132,94,105)(25,121,95,114)(26,130,96,103)(27,139,97,112)(28,128,98,101)(29,137,99,110)(30,126,100,119)(31,135,81,108)(32,124,82,117)(33,133,83,106)(34,122,84,115)(35,131,85,104)(36,140,86,113)(37,129,87,102)(38,138,88,111)(39,127,89,120)(40,136,90,109), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,35)(22,24)(23,33)(25,31)(26,40)(27,29)(28,38)(30,36)(32,34)(37,39)(41,57)(42,46)(43,55)(45,53)(47,51)(48,60)(50,58)(52,56)(61,158)(62,147)(63,156)(64,145)(65,154)(66,143)(67,152)(68,141)(69,150)(70,159)(71,148)(72,157)(73,146)(74,155)(75,144)(76,153)(77,142)(78,151)(79,160)(80,149)(81,95)(82,84)(83,93)(85,91)(86,100)(87,89)(88,98)(90,96)(92,94)(97,99)(101,138)(102,127)(103,136)(104,125)(105,134)(106,123)(107,132)(108,121)(109,130)(110,139)(111,128)(112,137)(113,126)(114,135)(115,124)(116,133)(117,122)(118,131)(119,140)(120,129)>;
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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,40)(19,39)(20,38)(41,85)(42,84)(43,83)(44,82)(45,81)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,120)(78,119)(79,118)(80,117)(121,148)(122,147)(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149), (1,152,59,67)(2,141,60,76)(3,150,41,65)(4,159,42,74)(5,148,43,63)(6,157,44,72)(7,146,45,61)(8,155,46,70)(9,144,47,79)(10,153,48,68)(11,142,49,77)(12,151,50,66)(13,160,51,75)(14,149,52,64)(15,158,53,73)(16,147,54,62)(17,156,55,71)(18,145,56,80)(19,154,57,69)(20,143,58,78)(21,125,91,118)(22,134,92,107)(23,123,93,116)(24,132,94,105)(25,121,95,114)(26,130,96,103)(27,139,97,112)(28,128,98,101)(29,137,99,110)(30,126,100,119)(31,135,81,108)(32,124,82,117)(33,133,83,106)(34,122,84,115)(35,131,85,104)(36,140,86,113)(37,129,87,102)(38,138,88,111)(39,127,89,120)(40,136,90,109), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,35)(22,24)(23,33)(25,31)(26,40)(27,29)(28,38)(30,36)(32,34)(37,39)(41,57)(42,46)(43,55)(45,53)(47,51)(48,60)(50,58)(52,56)(61,158)(62,147)(63,156)(64,145)(65,154)(66,143)(67,152)(68,141)(69,150)(70,159)(71,148)(72,157)(73,146)(74,155)(75,144)(76,153)(77,142)(78,151)(79,160)(80,149)(81,95)(82,84)(83,93)(85,91)(86,100)(87,89)(88,98)(90,96)(92,94)(97,99)(101,138)(102,127)(103,136)(104,125)(105,134)(106,123)(107,132)(108,121)(109,130)(110,139)(111,128)(112,137)(113,126)(114,135)(115,124)(116,133)(117,122)(118,131)(119,140)(120,129) );
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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,40),(19,39),(20,38),(41,85),(42,84),(43,83),(44,82),(45,81),(46,100),(47,99),(48,98),(49,97),(50,96),(51,95),(52,94),(53,93),(54,92),(55,91),(56,90),(57,89),(58,88),(59,87),(60,86),(61,116),(62,115),(63,114),(64,113),(65,112),(66,111),(67,110),(68,109),(69,108),(70,107),(71,106),(72,105),(73,104),(74,103),(75,102),(76,101),(77,120),(78,119),(79,118),(80,117),(121,148),(122,147),(123,146),(124,145),(125,144),(126,143),(127,142),(128,141),(129,160),(130,159),(131,158),(132,157),(133,156),(134,155),(135,154),(136,153),(137,152),(138,151),(139,150),(140,149)], [(1,152,59,67),(2,141,60,76),(3,150,41,65),(4,159,42,74),(5,148,43,63),(6,157,44,72),(7,146,45,61),(8,155,46,70),(9,144,47,79),(10,153,48,68),(11,142,49,77),(12,151,50,66),(13,160,51,75),(14,149,52,64),(15,158,53,73),(16,147,54,62),(17,156,55,71),(18,145,56,80),(19,154,57,69),(20,143,58,78),(21,125,91,118),(22,134,92,107),(23,123,93,116),(24,132,94,105),(25,121,95,114),(26,130,96,103),(27,139,97,112),(28,128,98,101),(29,137,99,110),(30,126,100,119),(31,135,81,108),(32,124,82,117),(33,133,83,106),(34,122,84,115),(35,131,85,104),(36,140,86,113),(37,129,87,102),(38,138,88,111),(39,127,89,120),(40,136,90,109)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,35),(22,24),(23,33),(25,31),(26,40),(27,29),(28,38),(30,36),(32,34),(37,39),(41,57),(42,46),(43,55),(45,53),(47,51),(48,60),(50,58),(52,56),(61,158),(62,147),(63,156),(64,145),(65,154),(66,143),(67,152),(68,141),(69,150),(70,159),(71,148),(72,157),(73,146),(74,155),(75,144),(76,153),(77,142),(78,151),(79,160),(80,149),(81,95),(82,84),(83,93),(85,91),(86,100),(87,89),(88,98),(90,96),(92,94),(97,99),(101,138),(102,127),(103,136),(104,125),(105,134),(106,123),(107,132),(108,121),(109,130),(110,139),(111,128),(112,137),(113,126),(114,135),(115,124),(116,133),(117,122),(118,131),(119,140),(120,129)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4oD4 | D10 | D10 | D10 | D10 | 2- 1+4 | D4xD5 | Q8.10D10 | D5xC4oD4 |
| kernel | D20:22D4 | D10.12D4 | D10:D4 | D5xC4:C4 | D20:8C4 | D10.13D4 | D10:2Q8 | C4xC5:D4 | Dic5:Q8 | C5xC22:Q8 | C2xC4oD20 | C2xQ8:2D5 | D20 | C22:Q8 | Dic5 | C22:C4 | C4:C4 | C22xC4 | C2xQ8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 6 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of D20:22D4 ►in GL6(F41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 37 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 6 | 0 | 0 |
| 0 | 0 | 4 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 37 | 0 | 0 | 0 | 0 |
| 26 | 29 | 0 | 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 | 6 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 5 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,37,9,0,0,0,0,0,0,0,40,0,0,0,0,1,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,4,0,0,0,0,6,10,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,26,0,0,0,0,37,29,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,5,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;
D20:22D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{22}D_4 % in TeX
G:=Group("D20:22D4"); // GroupNames label
G:=SmallGroup(320,1303);
// by ID
G=gap.SmallGroup(320,1303);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^9,c*b*c^-1=a^8*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations