metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.62D10, C23.15Dic10, (C23×C4).4D5, (C2×C20).451D4, C10.114(C4×D4), (C23×C20).2C2, C23.D5⋊17C4, C23.37(C4×D5), C10.69C22≀C2, (C22×C10).25Q8, C5⋊6(C23.8Q8), (C22×C4).406D10, (C22×C10).191D4, C2.1(C24⋊2D5), C23.81(C5⋊D4), C10.67(C22⋊Q8), C2.4(C20.48D4), C22.61(C4○D20), C22⋊3(C10.D4), (C23×C10).97C22, C22.31(C2×Dic10), C23.301(C22×D5), C10.10C42⋊22C2, (C22×C20).482C22, (C22×C10).361C23, C10.67(C22.D4), C2.3(C23.23D10), (C22×Dic5).64C22, (C2×C10)⋊9(C4⋊C4), C10.78(C2×C4⋊C4), C2.28(C4×C5⋊D4), (C2×C10).43(C2×Q8), C22.149(C2×C4×D5), (C2×Dic5)⋊10(C2×C4), (C2×C10).547(C2×D4), C22.85(C2×C5⋊D4), (C2×C10).89(C4○D4), (C2×C10.D4)⋊15C2, (C2×C4).224(C5⋊D4), C2.20(C2×C10.D4), (C2×C23.D5).16C2, (C2×C10).243(C22×C4), (C22×C10).167(C2×C4), SmallGroup(320,837)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.62D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=b, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de9 >
Subgroups: 638 in 234 conjugacy classes, 87 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×10], C22 [×3], C22 [×8], C22 [×12], C5, C2×C4 [×4], C2×C4 [×26], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×10], C24, Dic5 [×6], C20 [×4], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×4], C2×C20 [×12], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.8Q8, C10.D4 [×4], C23.D5 [×4], C23.D5 [×2], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20 [×2], C22×C20 [×6], C23×C10, C10.10C42 [×2], C2×C10.D4 [×2], C2×C23.D5 [×2], C23×C20, C24.62D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], D10 [×3], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×6], C22×D5, C23.8Q8, C10.D4 [×4], C2×Dic10, C2×C4×D5, C4○D20 [×2], C2×C5⋊D4 [×3], C2×C10.D4, C20.48D4 [×2], C4×C5⋊D4 [×2], C23.23D10, C24⋊2D5, C24.62D10
►On 160 points
Generators in S160
(1 136)(2 137)(3 138)(4 139)(5 140)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)(101 149)(102 150)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 150)(2 151)(3 152)(4 153)(5 154)(6 155)(7 156)(8 157)(9 158)(10 159)(11 160)(12 141)(13 142)(14 143)(15 144)(16 145)(17 146)(18 147)(19 148)(20 149)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(60 89)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)
(1 136)(2 137)(3 138)(4 139)(5 140)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 60)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)(73 95)(74 96)(75 97)(76 98)(77 99)(78 100)(79 81)(80 82)(101 149)(102 150)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)
(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 25 11 35)(2 53 12 43)(3 23 13 33)(4 51 14 41)(5 21 15 31)(6 49 16 59)(7 39 17 29)(8 47 18 57)(9 37 19 27)(10 45 20 55)(22 139 32 129)(24 137 34 127)(26 135 36 125)(28 133 38 123)(30 131 40 121)(42 128 52 138)(44 126 54 136)(46 124 56 134)(48 122 58 132)(50 140 60 130)(61 150 71 160)(62 111 72 101)(63 148 73 158)(64 109 74 119)(65 146 75 156)(66 107 76 117)(67 144 77 154)(68 105 78 115)(69 142 79 152)(70 103 80 113)(81 104 91 114)(82 141 92 151)(83 102 93 112)(84 159 94 149)(85 120 95 110)(86 157 96 147)(87 118 97 108)(88 155 98 145)(89 116 99 106)(90 153 100 143)
G:=sub<Sym(160)| (1,136)(2,137)(3,138)(4,139)(5,140)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,150)(2,151)(3,152)(4,153)(5,154)(6,155)(7,156)(8,157)(9,158)(10,159)(11,160)(12,141)(13,142)(14,143)(15,144)(16,145)(17,146)(18,147)(19,148)(20,149)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134), (1,136)(2,137)(3,138)(4,139)(5,140)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,60)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,99)(78,100)(79,81)(80,82)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148), (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), (1,25,11,35)(2,53,12,43)(3,23,13,33)(4,51,14,41)(5,21,15,31)(6,49,16,59)(7,39,17,29)(8,47,18,57)(9,37,19,27)(10,45,20,55)(22,139,32,129)(24,137,34,127)(26,135,36,125)(28,133,38,123)(30,131,40,121)(42,128,52,138)(44,126,54,136)(46,124,56,134)(48,122,58,132)(50,140,60,130)(61,150,71,160)(62,111,72,101)(63,148,73,158)(64,109,74,119)(65,146,75,156)(66,107,76,117)(67,144,77,154)(68,105,78,115)(69,142,79,152)(70,103,80,113)(81,104,91,114)(82,141,92,151)(83,102,93,112)(84,159,94,149)(85,120,95,110)(86,157,96,147)(87,118,97,108)(88,155,98,145)(89,116,99,106)(90,153,100,143)>;
G:=Group( (1,136)(2,137)(3,138)(4,139)(5,140)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,150)(2,151)(3,152)(4,153)(5,154)(6,155)(7,156)(8,157)(9,158)(10,159)(11,160)(12,141)(13,142)(14,143)(15,144)(16,145)(17,146)(18,147)(19,148)(20,149)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134), (1,136)(2,137)(3,138)(4,139)(5,140)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,60)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,99)(78,100)(79,81)(80,82)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148), (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), (1,25,11,35)(2,53,12,43)(3,23,13,33)(4,51,14,41)(5,21,15,31)(6,49,16,59)(7,39,17,29)(8,47,18,57)(9,37,19,27)(10,45,20,55)(22,139,32,129)(24,137,34,127)(26,135,36,125)(28,133,38,123)(30,131,40,121)(42,128,52,138)(44,126,54,136)(46,124,56,134)(48,122,58,132)(50,140,60,130)(61,150,71,160)(62,111,72,101)(63,148,73,158)(64,109,74,119)(65,146,75,156)(66,107,76,117)(67,144,77,154)(68,105,78,115)(69,142,79,152)(70,103,80,113)(81,104,91,114)(82,141,92,151)(83,102,93,112)(84,159,94,149)(85,120,95,110)(86,157,96,147)(87,118,97,108)(88,155,98,145)(89,116,99,106)(90,153,100,143) );
G=PermutationGroup([(1,136),(2,137),(3,138),(4,139),(5,140),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,127),(13,128),(14,129),(15,130),(16,131),(17,132),(18,133),(19,134),(20,135),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,97),(30,98),(31,99),(32,100),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78),(52,79),(53,80),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67),(101,149),(102,150),(103,151),(104,152),(105,153),(106,154),(107,155),(108,156),(109,157),(110,158),(111,159),(112,160),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)], [(1,150),(2,151),(3,152),(4,153),(5,154),(6,155),(7,156),(8,157),(9,158),(10,159),(11,160),(12,141),(13,142),(14,143),(15,144),(16,145),(17,146),(18,147),(19,148),(20,149),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,90),(42,91),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,99),(51,100),(52,81),(53,82),(54,83),(55,84),(56,85),(57,86),(58,87),(59,88),(60,89),(101,135),(102,136),(103,137),(104,138),(105,139),(106,140),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134)], [(1,136),(2,137),(3,138),(4,139),(5,140),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,127),(13,128),(14,129),(15,130),(16,131),(17,132),(18,133),(19,134),(20,135),(21,60),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(61,83),(62,84),(63,85),(64,86),(65,87),(66,88),(67,89),(68,90),(69,91),(70,92),(71,93),(72,94),(73,95),(74,96),(75,97),(76,98),(77,99),(78,100),(79,81),(80,82),(101,149),(102,150),(103,151),(104,152),(105,153),(106,154),(107,155),(108,156),(109,157),(110,158),(111,159),(112,160),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148)], [(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)], [(1,25,11,35),(2,53,12,43),(3,23,13,33),(4,51,14,41),(5,21,15,31),(6,49,16,59),(7,39,17,29),(8,47,18,57),(9,37,19,27),(10,45,20,55),(22,139,32,129),(24,137,34,127),(26,135,36,125),(28,133,38,123),(30,131,40,121),(42,128,52,138),(44,126,54,136),(46,124,56,134),(48,122,58,132),(50,140,60,130),(61,150,71,160),(62,111,72,101),(63,148,73,158),(64,109,74,119),(65,146,75,156),(66,107,76,117),(67,144,77,154),(68,105,78,115),(69,142,79,152),(70,103,80,113),(81,104,91,114),(82,141,92,151),(83,102,93,112),(84,159,94,149),(85,120,95,110),(86,157,96,147),(87,118,97,108),(88,155,98,145),(89,116,99,106),(90,153,100,143)])
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | Dic10 | C4×D5 | C5⋊D4 | C4○D20 |
| kernel | C24.62D10 | C10.10C42 | C2×C10.D4 | C2×C23.D5 | C23×C20 | C23.D5 | C2×C20 | C22×C10 | C22×C10 | C23×C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C23 | C23 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 16 | 8 | 8 | 8 | 16 |
Matrix representation of C24.62D10 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 33 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 |
| 0 | 0 | 0 | 10 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[32,0,0,0,0,0,5,0,0,0,0,0,33,0,0,0,0,0,10,0,0,0,0,0,4],[32,0,0,0,0,0,0,5,0,0,0,8,0,0,0,0,0,0,0,10,0,0,0,37,0] >;
C24.62D10 in GAP, Magma, Sage, TeX
C_2^4._{62}D_{10} % in TeX
G:=Group("C2^4.62D10"); // GroupNames label
G:=SmallGroup(320,837);
// by ID
G=gap.SmallGroup(320,837);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,758,58,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=b,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^9>;
// generators/relations