metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.8D10, C2.5(D4×Dic5), C10.117(C4×D4), C22⋊C4⋊5Dic5, C22.99(D4×D5), C2.5(D10⋊D4), C10.30(C4⋊D4), C23.7(C2×Dic5), (C2×Dic5).151D4, (C22×C4).307D10, C10.32(C4.4D4), C22.52(C4○D20), (C23×C10).34C22, C5⋊7(C24.C22), C23.281(C22×D5), C10.10C42⋊31C2, C10.63(C42⋊C2), C10.14(C42⋊2C2), C2.6(D10.12D4), C22.47(D4⋊2D5), (C22×C20).343C22, (C22×C10).326C23, C2.6(Dic5.5D4), C2.7(C23.D10), C22.40(C22×Dic5), C10.31(C22.D4), C2.8(C23.21D10), (C22×Dic5).208C22, (C2×C4×Dic5)⋊23C2, (C5×C22⋊C4)⋊14C4, (C2×C4⋊Dic5)⋊11C2, (C2×C20).333(C2×C4), (C2×C10).320(C2×D4), (C2×C22⋊C4).14D5, (C2×C4).16(C2×Dic5), (C2×C10).79(C4○D4), (C10×C22⋊C4).19C2, (C2×C23.D5).13C2, (C2×C10).280(C22×C4), (C22×C10).119(C2×C4), SmallGroup(320,578)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.8D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=c, f2=cb=bc, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 590 in 190 conjugacy classes, 75 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C24.C22, C4×Dic5, C4⋊Dic5, C23.D5, C5×C22⋊C4, C22×Dic5, C22×C20, C23×C10, C10.10C42, C2×C4×Dic5, C2×C4⋊Dic5, C2×C23.D5, C10×C22⋊C4, C24.8D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, Dic5, D10, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×Dic5, C22×D5, C24.C22, C4○D20, D4×D5, D4⋊2D5, C22×Dic5, C23.D10, D10.12D4, D10⋊D4, Dic5.5D4, C23.21D10, D4×Dic5, C24.8D10
►On 160 points
Generators in S160
(2 140)(4 122)(6 124)(8 126)(10 128)(12 130)(14 132)(16 134)(18 136)(20 138)(21 82)(22 32)(23 84)(24 34)(25 86)(26 36)(27 88)(28 38)(29 90)(30 40)(31 92)(33 94)(35 96)(37 98)(39 100)(41 51)(42 115)(43 53)(44 117)(45 55)(46 119)(47 57)(48 101)(49 59)(50 103)(52 105)(54 107)(56 109)(58 111)(60 113)(61 146)(63 148)(65 150)(67 152)(69 154)(71 156)(73 158)(75 160)(77 142)(79 144)(81 91)(83 93)(85 95)(87 97)(89 99)(102 112)(104 114)(106 116)(108 118)(110 120)
(1 151)(2 152)(3 153)(4 154)(5 155)(6 156)(7 157)(8 158)(9 159)(10 160)(11 141)(12 142)(13 143)(14 144)(15 145)(16 146)(17 147)(18 148)(19 149)(20 150)(21 56)(22 57)(23 58)(24 59)(25 60)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 129)(77 130)(78 131)(79 132)(80 133)(81 108)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)
(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)
(1 139)(2 140)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 101)(59 102)(60 103)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 141)(77 142)(78 143)(79 144)(80 145)
(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 23 141 48)(2 32 142 57)(3 21 143 46)(4 30 144 55)(5 39 145 44)(6 28 146 53)(7 37 147 42)(8 26 148 51)(9 35 149 60)(10 24 150 49)(11 33 151 58)(12 22 152 47)(13 31 153 56)(14 40 154 45)(15 29 155 54)(16 38 156 43)(17 27 157 52)(18 36 158 41)(19 25 159 50)(20 34 160 59)(61 116 124 99)(62 105 125 88)(63 114 126 97)(64 103 127 86)(65 112 128 95)(66 101 129 84)(67 110 130 93)(68 119 131 82)(69 108 132 91)(70 117 133 100)(71 106 134 89)(72 115 135 98)(73 104 136 87)(74 113 137 96)(75 102 138 85)(76 111 139 94)(77 120 140 83)(78 109 121 92)(79 118 122 81)(80 107 123 90)
G:=sub<Sym(160)| (2,140)(4,122)(6,124)(8,126)(10,128)(12,130)(14,132)(16,134)(18,136)(20,138)(21,82)(22,32)(23,84)(24,34)(25,86)(26,36)(27,88)(28,38)(29,90)(30,40)(31,92)(33,94)(35,96)(37,98)(39,100)(41,51)(42,115)(43,53)(44,117)(45,55)(46,119)(47,57)(48,101)(49,59)(50,103)(52,105)(54,107)(56,109)(58,111)(60,113)(61,146)(63,148)(65,150)(67,152)(69,154)(71,156)(73,158)(75,160)(77,142)(79,144)(81,91)(83,93)(85,95)(87,97)(89,99)(102,112)(104,114)(106,116)(108,118)(110,120), (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,159)(10,160)(11,141)(12,142)(13,143)(14,144)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,56)(22,57)(23,58)(24,59)(25,60)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133)(81,108)(82,109)(83,110)(84,111)(85,112)(86,113)(87,114)(88,115)(89,116)(90,117)(91,118)(92,119)(93,120)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107), (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), (1,139)(2,140)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,101)(59,102)(60,103)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,141)(77,142)(78,143)(79,144)(80,145), (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), (1,23,141,48)(2,32,142,57)(3,21,143,46)(4,30,144,55)(5,39,145,44)(6,28,146,53)(7,37,147,42)(8,26,148,51)(9,35,149,60)(10,24,150,49)(11,33,151,58)(12,22,152,47)(13,31,153,56)(14,40,154,45)(15,29,155,54)(16,38,156,43)(17,27,157,52)(18,36,158,41)(19,25,159,50)(20,34,160,59)(61,116,124,99)(62,105,125,88)(63,114,126,97)(64,103,127,86)(65,112,128,95)(66,101,129,84)(67,110,130,93)(68,119,131,82)(69,108,132,91)(70,117,133,100)(71,106,134,89)(72,115,135,98)(73,104,136,87)(74,113,137,96)(75,102,138,85)(76,111,139,94)(77,120,140,83)(78,109,121,92)(79,118,122,81)(80,107,123,90)>;
G:=Group( (2,140)(4,122)(6,124)(8,126)(10,128)(12,130)(14,132)(16,134)(18,136)(20,138)(21,82)(22,32)(23,84)(24,34)(25,86)(26,36)(27,88)(28,38)(29,90)(30,40)(31,92)(33,94)(35,96)(37,98)(39,100)(41,51)(42,115)(43,53)(44,117)(45,55)(46,119)(47,57)(48,101)(49,59)(50,103)(52,105)(54,107)(56,109)(58,111)(60,113)(61,146)(63,148)(65,150)(67,152)(69,154)(71,156)(73,158)(75,160)(77,142)(79,144)(81,91)(83,93)(85,95)(87,97)(89,99)(102,112)(104,114)(106,116)(108,118)(110,120), (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,159)(10,160)(11,141)(12,142)(13,143)(14,144)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,56)(22,57)(23,58)(24,59)(25,60)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133)(81,108)(82,109)(83,110)(84,111)(85,112)(86,113)(87,114)(88,115)(89,116)(90,117)(91,118)(92,119)(93,120)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107), (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), (1,139)(2,140)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,101)(59,102)(60,103)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,141)(77,142)(78,143)(79,144)(80,145), (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), (1,23,141,48)(2,32,142,57)(3,21,143,46)(4,30,144,55)(5,39,145,44)(6,28,146,53)(7,37,147,42)(8,26,148,51)(9,35,149,60)(10,24,150,49)(11,33,151,58)(12,22,152,47)(13,31,153,56)(14,40,154,45)(15,29,155,54)(16,38,156,43)(17,27,157,52)(18,36,158,41)(19,25,159,50)(20,34,160,59)(61,116,124,99)(62,105,125,88)(63,114,126,97)(64,103,127,86)(65,112,128,95)(66,101,129,84)(67,110,130,93)(68,119,131,82)(69,108,132,91)(70,117,133,100)(71,106,134,89)(72,115,135,98)(73,104,136,87)(74,113,137,96)(75,102,138,85)(76,111,139,94)(77,120,140,83)(78,109,121,92)(79,118,122,81)(80,107,123,90) );
G=PermutationGroup([[(2,140),(4,122),(6,124),(8,126),(10,128),(12,130),(14,132),(16,134),(18,136),(20,138),(21,82),(22,32),(23,84),(24,34),(25,86),(26,36),(27,88),(28,38),(29,90),(30,40),(31,92),(33,94),(35,96),(37,98),(39,100),(41,51),(42,115),(43,53),(44,117),(45,55),(46,119),(47,57),(48,101),(49,59),(50,103),(52,105),(54,107),(56,109),(58,111),(60,113),(61,146),(63,148),(65,150),(67,152),(69,154),(71,156),(73,158),(75,160),(77,142),(79,144),(81,91),(83,93),(85,95),(87,97),(89,99),(102,112),(104,114),(106,116),(108,118),(110,120)], [(1,151),(2,152),(3,153),(4,154),(5,155),(6,156),(7,157),(8,158),(9,159),(10,160),(11,141),(12,142),(13,143),(14,144),(15,145),(16,146),(17,147),(18,148),(19,149),(20,150),(21,56),(22,57),(23,58),(24,59),(25,60),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,121),(69,122),(70,123),(71,124),(72,125),(73,126),(74,127),(75,128),(76,129),(77,130),(78,131),(79,132),(80,133),(81,108),(82,109),(83,110),(84,111),(85,112),(86,113),(87,114),(88,115),(89,116),(90,117),(91,118),(92,119),(93,120),(94,101),(95,102),(96,103),(97,104),(98,105),(99,106),(100,107)], [(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)], [(1,139),(2,140),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,133),(16,134),(17,135),(18,136),(19,137),(20,138),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,81),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,101),(59,102),(60,103),(61,146),(62,147),(63,148),(64,149),(65,150),(66,151),(67,152),(68,153),(69,154),(70,155),(71,156),(72,157),(73,158),(74,159),(75,160),(76,141),(77,142),(78,143),(79,144),(80,145)], [(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)], [(1,23,141,48),(2,32,142,57),(3,21,143,46),(4,30,144,55),(5,39,145,44),(6,28,146,53),(7,37,147,42),(8,26,148,51),(9,35,149,60),(10,24,150,49),(11,33,151,58),(12,22,152,47),(13,31,153,56),(14,40,154,45),(15,29,155,54),(16,38,156,43),(17,27,157,52),(18,36,158,41),(19,25,159,50),(20,34,160,59),(61,116,124,99),(62,105,125,88),(63,114,126,97),(64,103,127,86),(65,112,128,95),(66,101,129,84),(67,110,130,93),(68,119,131,82),(69,108,132,91),(70,117,133,100),(71,106,134,89),(72,115,135,98),(73,104,136,87),(74,113,137,96),(75,102,138,85),(76,111,139,94),(77,120,140,83),(78,109,121,92),(79,118,122,81),(80,107,123,90)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | Dic5 | D10 | D10 | C4○D20 | D4×D5 | D4⋊2D5 |
| kernel | C24.8D10 | C10.10C42 | C2×C4×Dic5 | C2×C4⋊Dic5 | C2×C23.D5 | C10×C22⋊C4 | C5×C22⋊C4 | C2×Dic5 | C2×C22⋊C4 | C2×C10 | C22⋊C4 | C22×C4 | C24 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 4 | 2 | 8 | 8 | 4 | 2 | 16 | 4 | 4 |
Matrix representation of C24.8D10 ►in GL6(𝔽41)
| 1 | 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 | 31 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 1 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 |
| 0 | 0 | 0 | 0 | 8 | 39 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 3 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 38 |
| 0 | 0 | 0 | 0 | 20 | 15 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,31,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,35,0,0,0,0,7,6,0,0,0,0,0,0,21,8,0,0,0,0,0,39],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,21,3,0,0,0,0,3,20,0,0,0,0,0,0,26,20,0,0,0,0,38,15] >;
C24.8D10 in GAP, Magma, Sage, TeX
C_2^4._8D_{10} % in TeX
G:=Group("C2^4.8D10"); // GroupNames label
G:=SmallGroup(320,578);
// by ID
G=gap.SmallGroup(320,578);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=c,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,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=e^9>;
// generators/relations