metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.16D10, C23.16D20, (C2×C20)⋊7D4, C10.34C22≀C2, C2.7(C20⋊2D4), C2.8(C20⋊7D4), (C2×Dic5).65D4, (C22×D5).40D4, C22.243(D4×D5), (C22×C4).36D10, (C22×C10).70D4, C10.60(C4⋊D4), C22.127(C2×D20), C5⋊3(C23.10D4), C2.35(C22⋊D20), C10.36(C4.4D4), (C22×C20).62C22, (C23×C10).44C22, (C23×D5).17C22, C23.373(C22×D5), C10.10C42⋊18C2, C2.11(Dic5⋊D4), C22.101(C4○D20), C22.98(D4⋊2D5), (C22×C10).335C23, C2.23(D10.12D4), C2.23(Dic5.5D4), C10.35(C22.D4), C2.17(C22.D20), (C22×Dic5).47C22, (C2×C4)⋊4(C5⋊D4), (C2×C22⋊C4)⋊9D5, (C2×C4⋊Dic5)⋊13C2, (C2×C23.D5)⋊6C2, (C2×D10⋊C4)⋊9C2, (C10×C22⋊C4)⋊12C2, (C2×C10).326(C2×D4), (C22×C5⋊D4).6C2, (C2×C10).81(C4○D4), C22.129(C2×C5⋊D4), SmallGroup(320,588)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.16D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=dc=cd, f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >
Subgroups: 998 in 238 conjugacy classes, 63 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C23.10D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C22×C20, C23×D5, C23×C10, C10.10C42, C2×C4⋊Dic5, C2×D10⋊C4, C2×C23.D5, C10×C22⋊C4, C22×C5⋊D4, C24.16D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22≀C2, C4⋊D4, C22.D4, C4.4D4, D20, C5⋊D4, C22×D5, C23.10D4, C2×D20, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, C22⋊D20, D10.12D4, Dic5.5D4, C22.D20, C20⋊7D4, C20⋊2D4, Dic5⋊D4, C24.16D10
►On 160 points
Generators in S160
(1 104)(2 131)(3 106)(4 133)(5 108)(6 135)(7 110)(8 137)(9 112)(10 139)(11 114)(12 121)(13 116)(14 123)(15 118)(16 125)(17 120)(18 127)(19 102)(20 129)(21 156)(22 65)(23 158)(24 67)(25 160)(26 69)(27 142)(28 71)(29 144)(30 73)(31 146)(32 75)(33 148)(34 77)(35 150)(36 79)(37 152)(38 61)(39 154)(40 63)(41 119)(42 126)(43 101)(44 128)(45 103)(46 130)(47 105)(48 132)(49 107)(50 134)(51 109)(52 136)(53 111)(54 138)(55 113)(56 140)(57 115)(58 122)(59 117)(60 124)(62 94)(64 96)(66 98)(68 100)(70 82)(72 84)(74 86)(76 88)(78 90)(80 92)(81 141)(83 143)(85 145)(87 147)(89 149)(91 151)(93 153)(95 155)(97 157)(99 159)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 21)(19 22)(20 23)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 89)(57 90)(58 91)(59 92)(60 93)(61 118)(62 119)(63 120)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 81)(37 82)(38 83)(39 84)(40 85)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 155)(74 156)(75 157)(76 158)(77 159)(78 160)(79 141)(80 142)(101 137)(102 138)(103 139)(104 140)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)(111 127)(112 128)(113 129)(114 130)(115 131)(116 132)(117 133)(118 134)(119 135)(120 136)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 41)(17 42)(18 43)(19 44)(20 45)(21 96)(22 97)(23 98)(24 99)(25 100)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 141)(70 142)(71 143)(72 144)(73 145)(74 146)(75 147)(76 148)(77 149)(78 150)(79 151)(80 152)(101 127)(102 128)(103 129)(104 130)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)(113 139)(114 140)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)
(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 10 46 55)(2 54 47 9)(3 8 48 53)(4 52 49 7)(5 6 50 51)(11 20 56 45)(12 44 57 19)(13 18 58 43)(14 42 59 17)(15 16 60 41)(21 91 96 36)(22 35 97 90)(23 89 98 34)(24 33 99 88)(25 87 100 32)(26 31 81 86)(27 85 82 30)(28 29 83 84)(37 95 92 40)(38 39 93 94)(61 119 153 125)(62 124 154 118)(63 117 155 123)(64 122 156 116)(65 115 157 121)(66 140 158 114)(67 113 159 139)(68 138 160 112)(69 111 141 137)(70 136 142 110)(71 109 143 135)(72 134 144 108)(73 107 145 133)(74 132 146 106)(75 105 147 131)(76 130 148 104)(77 103 149 129)(78 128 150 102)(79 101 151 127)(80 126 152 120)
G:=sub<Sym(160)| (1,104)(2,131)(3,106)(4,133)(5,108)(6,135)(7,110)(8,137)(9,112)(10,139)(11,114)(12,121)(13,116)(14,123)(15,118)(16,125)(17,120)(18,127)(19,102)(20,129)(21,156)(22,65)(23,158)(24,67)(25,160)(26,69)(27,142)(28,71)(29,144)(30,73)(31,146)(32,75)(33,148)(34,77)(35,150)(36,79)(37,152)(38,61)(39,154)(40,63)(41,119)(42,126)(43,101)(44,128)(45,103)(46,130)(47,105)(48,132)(49,107)(50,134)(51,109)(52,136)(53,111)(54,138)(55,113)(56,140)(57,115)(58,122)(59,117)(60,124)(62,94)(64,96)(66,98)(68,100)(70,82)(72,84)(74,86)(76,88)(78,90)(80,92)(81,141)(83,143)(85,145)(87,147)(89,149)(91,151)(93,153)(95,155)(97,157)(99,159), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,89)(57,90)(58,91)(59,92)(60,93)(61,118)(62,119)(63,120)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149), (1,56)(2,57)(3,58)(4,59)(5,60)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,81)(37,82)(38,83)(39,84)(40,85)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,141)(80,142)(101,137)(102,138)(103,139)(104,140)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,41)(17,42)(18,43)(19,44)(20,45)(21,96)(22,97)(23,98)(24,99)(25,100)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,141)(70,142)(71,143)(72,144)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (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), (1,10,46,55)(2,54,47,9)(3,8,48,53)(4,52,49,7)(5,6,50,51)(11,20,56,45)(12,44,57,19)(13,18,58,43)(14,42,59,17)(15,16,60,41)(21,91,96,36)(22,35,97,90)(23,89,98,34)(24,33,99,88)(25,87,100,32)(26,31,81,86)(27,85,82,30)(28,29,83,84)(37,95,92,40)(38,39,93,94)(61,119,153,125)(62,124,154,118)(63,117,155,123)(64,122,156,116)(65,115,157,121)(66,140,158,114)(67,113,159,139)(68,138,160,112)(69,111,141,137)(70,136,142,110)(71,109,143,135)(72,134,144,108)(73,107,145,133)(74,132,146,106)(75,105,147,131)(76,130,148,104)(77,103,149,129)(78,128,150,102)(79,101,151,127)(80,126,152,120)>;
G:=Group( (1,104)(2,131)(3,106)(4,133)(5,108)(6,135)(7,110)(8,137)(9,112)(10,139)(11,114)(12,121)(13,116)(14,123)(15,118)(16,125)(17,120)(18,127)(19,102)(20,129)(21,156)(22,65)(23,158)(24,67)(25,160)(26,69)(27,142)(28,71)(29,144)(30,73)(31,146)(32,75)(33,148)(34,77)(35,150)(36,79)(37,152)(38,61)(39,154)(40,63)(41,119)(42,126)(43,101)(44,128)(45,103)(46,130)(47,105)(48,132)(49,107)(50,134)(51,109)(52,136)(53,111)(54,138)(55,113)(56,140)(57,115)(58,122)(59,117)(60,124)(62,94)(64,96)(66,98)(68,100)(70,82)(72,84)(74,86)(76,88)(78,90)(80,92)(81,141)(83,143)(85,145)(87,147)(89,149)(91,151)(93,153)(95,155)(97,157)(99,159), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,89)(57,90)(58,91)(59,92)(60,93)(61,118)(62,119)(63,120)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149), (1,56)(2,57)(3,58)(4,59)(5,60)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,81)(37,82)(38,83)(39,84)(40,85)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,141)(80,142)(101,137)(102,138)(103,139)(104,140)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,41)(17,42)(18,43)(19,44)(20,45)(21,96)(22,97)(23,98)(24,99)(25,100)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,141)(70,142)(71,143)(72,144)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (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), (1,10,46,55)(2,54,47,9)(3,8,48,53)(4,52,49,7)(5,6,50,51)(11,20,56,45)(12,44,57,19)(13,18,58,43)(14,42,59,17)(15,16,60,41)(21,91,96,36)(22,35,97,90)(23,89,98,34)(24,33,99,88)(25,87,100,32)(26,31,81,86)(27,85,82,30)(28,29,83,84)(37,95,92,40)(38,39,93,94)(61,119,153,125)(62,124,154,118)(63,117,155,123)(64,122,156,116)(65,115,157,121)(66,140,158,114)(67,113,159,139)(68,138,160,112)(69,111,141,137)(70,136,142,110)(71,109,143,135)(72,134,144,108)(73,107,145,133)(74,132,146,106)(75,105,147,131)(76,130,148,104)(77,103,149,129)(78,128,150,102)(79,101,151,127)(80,126,152,120) );
G=PermutationGroup([[(1,104),(2,131),(3,106),(4,133),(5,108),(6,135),(7,110),(8,137),(9,112),(10,139),(11,114),(12,121),(13,116),(14,123),(15,118),(16,125),(17,120),(18,127),(19,102),(20,129),(21,156),(22,65),(23,158),(24,67),(25,160),(26,69),(27,142),(28,71),(29,144),(30,73),(31,146),(32,75),(33,148),(34,77),(35,150),(36,79),(37,152),(38,61),(39,154),(40,63),(41,119),(42,126),(43,101),(44,128),(45,103),(46,130),(47,105),(48,132),(49,107),(50,134),(51,109),(52,136),(53,111),(54,138),(55,113),(56,140),(57,115),(58,122),(59,117),(60,124),(62,94),(64,96),(66,98),(68,100),(70,82),(72,84),(74,86),(76,88),(78,90),(80,92),(81,141),(83,143),(85,145),(87,147),(89,149),(91,151),(93,153),(95,155),(97,157),(99,159)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,21),(19,22),(20,23),(41,94),(42,95),(43,96),(44,97),(45,98),(46,99),(47,100),(48,81),(49,82),(50,83),(51,84),(52,85),(53,86),(54,87),(55,88),(56,89),(57,90),(58,91),(59,92),(60,93),(61,118),(62,119),(63,120),(64,101),(65,102),(66,103),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,113),(77,114),(78,115),(79,116),(80,117),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,157),(129,158),(130,159),(131,160),(132,141),(133,142),(134,143),(135,144),(136,145),(137,146),(138,147),(139,148),(140,149)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,86),(22,87),(23,88),(24,89),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,81),(37,82),(38,83),(39,84),(40,85),(61,143),(62,144),(63,145),(64,146),(65,147),(66,148),(67,149),(68,150),(69,151),(70,152),(71,153),(72,154),(73,155),(74,156),(75,157),(76,158),(77,159),(78,160),(79,141),(80,142),(101,137),(102,138),(103,139),(104,140),(105,121),(106,122),(107,123),(108,124),(109,125),(110,126),(111,127),(112,128),(113,129),(114,130),(115,131),(116,132),(117,133),(118,134),(119,135),(120,136)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,41),(17,42),(18,43),(19,44),(20,45),(21,96),(22,97),(23,98),(24,99),(25,100),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(61,153),(62,154),(63,155),(64,156),(65,157),(66,158),(67,159),(68,160),(69,141),(70,142),(71,143),(72,144),(73,145),(74,146),(75,147),(76,148),(77,149),(78,150),(79,151),(80,152),(101,127),(102,128),(103,129),(104,130),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,137),(112,138),(113,139),(114,140),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126)], [(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)], [(1,10,46,55),(2,54,47,9),(3,8,48,53),(4,52,49,7),(5,6,50,51),(11,20,56,45),(12,44,57,19),(13,18,58,43),(14,42,59,17),(15,16,60,41),(21,91,96,36),(22,35,97,90),(23,89,98,34),(24,33,99,88),(25,87,100,32),(26,31,81,86),(27,85,82,30),(28,29,83,84),(37,95,92,40),(38,39,93,94),(61,119,153,125),(62,124,154,118),(63,117,155,123),(64,122,156,116),(65,115,157,121),(66,140,158,114),(67,113,159,139),(68,138,160,112),(69,111,141,137),(70,136,142,110),(71,109,143,135),(72,134,144,108),(73,107,145,133),(74,132,146,106),(75,105,147,131),(76,130,148,104),(77,103,149,129),(78,128,150,102),(79,101,151,127),(80,126,152,120)]])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | C4○D20 | D4×D5 | D4⋊2D5 |
| kernel | C24.16D10 | C10.10C42 | C2×C4⋊Dic5 | C2×D10⋊C4 | C2×C23.D5 | C10×C22⋊C4 | C22×C5⋊D4 | C2×Dic5 | C2×C20 | C22×D5 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 4 | 2 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C24.16D10 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 35 | 0 | 0 |
| 0 | 0 | 6 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 35 |
| 0 | 0 | 0 | 0 | 6 | 23 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 25 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 2 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 1 | 35 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,23,6,0,0,0,0,35,18,0,0,0,0,0,0,18,6,0,0,0,0,35,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,2,16,0,0,0,0,25,16,0,0,0,0,0,0,6,35,0,0,0,0,6,1],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,25,16,0,0,0,0,2,16,0,0,0,0,0,0,6,1,0,0,0,0,6,35] >;
C24.16D10 in GAP, Magma, Sage, TeX
C_2^4._{16}D_{10} % in TeX
G:=Group("C2^4.16D10"); // GroupNames label
G:=SmallGroup(320,588);
// by ID
G=gap.SmallGroup(320,588);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=d*c=c*d,f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^9>;
// generators/relations