metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊17D4, Dic10⋊16D4, C4⋊D4⋊2D5, C4.99(D4×D5), C5⋊4(D4⋊D4), C4⋊C4.57D10, (C2×D4).37D10, C20.146(C2×D4), (C2×C20).262D4, D20⋊6C4⋊34C2, C10.45C22≀C2, C10.96(C4○D8), C10.Q16⋊33C2, (C22×C10).83D4, C20.55D4⋊11C2, C10.90(C8⋊C22), (C2×C20).356C23, (D4×C10).53C22, (C22×C4).120D10, C23.23(C5⋊D4), C2.13(C23⋊D10), (C2×D20).249C22, C2.15(D4.8D10), C2.11(D4.D10), (C22×C20).160C22, (C2×Dic10).276C22, (C2×D4⋊D5)⋊9C2, (C5×C4⋊D4)⋊2C2, (C2×D4.D5)⋊8C2, (C2×C4○D20)⋊15C2, (C2×C10).487(C2×D4), (C2×C4).171(C5⋊D4), (C5×C4⋊C4).104C22, (C2×C4).456(C22×D5), C22.162(C2×C5⋊D4), (C2×C5⋊2C8).108C22, SmallGroup(320,664)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊17D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=a11, ad=da, cbc-1=a5b, dbd=a10b, dcd=c-1 >
Subgroups: 670 in 162 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C5⋊2C8, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, D4⋊D4, C2×C5⋊2C8, D4⋊D5, D4.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, D20⋊6C4, C10.Q16, C20.55D4, C2×D4⋊D5, C2×D4.D5, C5×C4⋊D4, C2×C4○D20, D20⋊17D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8⋊C22, C5⋊D4, C22×D5, D4⋊D4, D4×D5, C2×C5⋊D4, D4.D10, C23⋊D10, D4.8D10, D20⋊17D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)(101 103)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(135 140)(136 139)(137 138)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)
(1 92 36 105)(2 83 37 116)(3 94 38 107)(4 85 39 118)(5 96 40 109)(6 87 21 120)(7 98 22 111)(8 89 23 102)(9 100 24 113)(10 91 25 104)(11 82 26 115)(12 93 27 106)(13 84 28 117)(14 95 29 108)(15 86 30 119)(16 97 31 110)(17 88 32 101)(18 99 33 112)(19 90 34 103)(20 81 35 114)(41 76 135 150)(42 67 136 141)(43 78 137 152)(44 69 138 143)(45 80 139 154)(46 71 140 145)(47 62 121 156)(48 73 122 147)(49 64 123 158)(50 75 124 149)(51 66 125 160)(52 77 126 151)(53 68 127 142)(54 79 128 153)(55 70 129 144)(56 61 130 155)(57 72 131 146)(58 63 132 157)(59 74 133 148)(60 65 134 159)
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 121)(20 122)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(40 53)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(101 154)(102 155)(103 156)(104 157)(105 158)(106 159)(107 160)(108 141)(109 142)(110 143)(111 144)(112 145)(113 146)(114 147)(115 148)(116 149)(117 150)(118 151)(119 152)(120 153)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,140)(136,139)(137,138)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151), (1,92,36,105)(2,83,37,116)(3,94,38,107)(4,85,39,118)(5,96,40,109)(6,87,21,120)(7,98,22,111)(8,89,23,102)(9,100,24,113)(10,91,25,104)(11,82,26,115)(12,93,27,106)(13,84,28,117)(14,95,29,108)(15,86,30,119)(16,97,31,110)(17,88,32,101)(18,99,33,112)(19,90,34,103)(20,81,35,114)(41,76,135,150)(42,67,136,141)(43,78,137,152)(44,69,138,143)(45,80,139,154)(46,71,140,145)(47,62,121,156)(48,73,122,147)(49,64,123,158)(50,75,124,149)(51,66,125,160)(52,77,126,151)(53,68,127,142)(54,79,128,153)(55,70,129,144)(56,61,130,155)(57,72,131,146)(58,63,132,157)(59,74,133,148)(60,65,134,159), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,121)(20,122)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(40,53)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,141)(109,142)(110,143)(111,144)(112,145)(113,146)(114,147)(115,148)(116,149)(117,150)(118,151)(119,152)(120,153)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,140)(136,139)(137,138)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151), (1,92,36,105)(2,83,37,116)(3,94,38,107)(4,85,39,118)(5,96,40,109)(6,87,21,120)(7,98,22,111)(8,89,23,102)(9,100,24,113)(10,91,25,104)(11,82,26,115)(12,93,27,106)(13,84,28,117)(14,95,29,108)(15,86,30,119)(16,97,31,110)(17,88,32,101)(18,99,33,112)(19,90,34,103)(20,81,35,114)(41,76,135,150)(42,67,136,141)(43,78,137,152)(44,69,138,143)(45,80,139,154)(46,71,140,145)(47,62,121,156)(48,73,122,147)(49,64,123,158)(50,75,124,149)(51,66,125,160)(52,77,126,151)(53,68,127,142)(54,79,128,153)(55,70,129,144)(56,61,130,155)(57,72,131,146)(58,63,132,157)(59,74,133,148)(60,65,134,159), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,121)(20,122)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(40,53)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,141)(109,142)(110,143)(111,144)(112,145)(113,146)(114,147)(115,148)(116,149)(117,150)(118,151)(119,152)(120,153) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,46),(42,45),(43,44),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(98,100),(101,103),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(121,134),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(135,140),(136,139),(137,138),(141,159),(142,158),(143,157),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151)], [(1,92,36,105),(2,83,37,116),(3,94,38,107),(4,85,39,118),(5,96,40,109),(6,87,21,120),(7,98,22,111),(8,89,23,102),(9,100,24,113),(10,91,25,104),(11,82,26,115),(12,93,27,106),(13,84,28,117),(14,95,29,108),(15,86,30,119),(16,97,31,110),(17,88,32,101),(18,99,33,112),(19,90,34,103),(20,81,35,114),(41,76,135,150),(42,67,136,141),(43,78,137,152),(44,69,138,143),(45,80,139,154),(46,71,140,145),(47,62,121,156),(48,73,122,147),(49,64,123,158),(50,75,124,149),(51,66,125,160),(52,77,126,151),(53,68,127,142),(54,79,128,153),(55,70,129,144),(56,61,130,155),(57,72,131,146),(58,63,132,157),(59,74,133,148),(60,65,134,159)], [(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,129),(8,130),(9,131),(10,132),(11,133),(12,134),(13,135),(14,136),(15,137),(16,138),(17,139),(18,140),(19,121),(20,122),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(40,53),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(101,154),(102,155),(103,156),(104,157),(105,158),(106,159),(107,160),(108,141),(109,142),(110,143),(111,144),(112,145),(113,146),(114,147),(115,148),(116,149),(117,150),(118,151),(119,152),(120,153)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 20 | 20 | 2 | 2 | 2 | 2 | 8 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4×D5 | D4.D10 | D4.8D10 |
| kernel | D20⋊17D4 | D20⋊6C4 | C10.Q16 | C20.55D4 | C2×D4⋊D5 | C2×D4.D5 | C5×C4⋊D4 | C2×C4○D20 | Dic10 | D20 | C2×C20 | C22×C10 | C4⋊D4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C2×C4 | C23 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of D20⋊17D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 5 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 0 | 37 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 39 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 32 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 22 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 16 |
| 0 | 0 | 0 | 0 | 36 | 9 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,1,37,0,0,0,0,21,40],[40,39,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,21,40],[9,0,0,0,0,0,32,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,22,0,0,0,0,13,0],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,36,0,0,0,0,16,9] >;
D20⋊17D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{17}D_4 % in TeX
G:=Group("D20:17D4"); // GroupNames label
G:=SmallGroup(320,664);
// by ID
G=gap.SmallGroup(320,664);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,1123,297,136,12550]);
// 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=a^11,a*d=d*a,c*b*c^-1=a^5*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations