metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.19D10, (D4×C10)⋊19C4, (C2×D4)⋊7Dic5, C20⋊6(C22⋊C4), (C2×Dic5)⋊12D4, C10.132(C4×D4), (C2×C20).191D4, C4⋊1(C23.D5), C2.19(D4×Dic5), (C22×D4).7D5, C2.4(C20⋊D4), C2.5(C20⋊2D4), C22.121(D4×D5), C10.36(C4⋊1D4), (C22×C4).353D10, C10.129(C4⋊D4), C23.10(C2×Dic5), C2.4(C20.17D4), C10.47(C4.4D4), (C23×C10).47C22, C5⋊5(C24.3C22), C23.306(C22×D5), C22.62(D4⋊2D5), (C22×C20).199C22, (C22×C10).366C23, C22.52(C22×Dic5), (C22×Dic5).220C22, (C2×C4×Dic5)⋊3C2, (D4×C2×C10).5C2, (C2×C4⋊Dic5)⋊35C2, (C2×C20).291(C2×C4), (C2×C10).555(C2×D4), (C2×C23.D5)⋊10C2, (C2×C4).50(C2×Dic5), C22.92(C2×C5⋊D4), C2.12(C2×C23.D5), (C2×C4).148(C5⋊D4), C10.117(C2×C22⋊C4), (C2×C10).162(C4○D4), (C22×C10).141(C2×C4), (C2×C10).298(C22×C4), SmallGroup(320,848)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.19D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=d, f2=db=bd, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 766 in 258 conjugacy classes, 91 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C24.3C22, C4×Dic5, C4⋊Dic5, C23.D5, C22×Dic5, C22×C20, D4×C10, D4×C10, C23×C10, C2×C4×Dic5, C2×C4⋊Dic5, C2×C23.D5, D4×C2×C10, C24.19D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C4○D4, Dic5, D10, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×Dic5, C5⋊D4, C22×D5, C24.3C22, C23.D5, D4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×Dic5, C20.17D4, C20⋊2D4, C20⋊D4, C2×C23.D5, C24.19D10
►On 160 points
Generators in S160
(1 16)(2 7)(3 18)(4 9)(5 20)(6 11)(8 13)(10 15)(12 17)(14 19)(21 54)(22 45)(23 56)(24 47)(25 58)(26 49)(27 60)(28 51)(29 42)(30 53)(31 44)(32 55)(33 46)(34 57)(35 48)(36 59)(37 50)(38 41)(39 52)(40 43)(61 66)(62 77)(63 68)(64 79)(65 70)(67 72)(69 74)(71 76)(73 78)(75 80)(81 96)(82 87)(83 98)(84 89)(85 100)(86 91)(88 93)(90 95)(92 97)(94 99)(101 106)(102 117)(103 108)(104 119)(105 110)(107 112)(109 114)(111 116)(113 118)(115 120)(121 147)(122 158)(123 149)(124 160)(125 151)(126 142)(127 153)(128 144)(129 155)(130 146)(131 157)(132 148)(133 159)(134 150)(135 141)(136 152)(137 143)(138 154)(139 145)(140 156)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 97)(16 98)(17 99)(18 100)(19 81)(20 82)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 160)(42 141)(43 142)(44 143)(45 144)(46 145)(47 146)(48 147)(49 148)(50 149)(51 150)(52 151)(53 152)(54 153)(55 154)(56 155)(57 156)(58 157)(59 158)(60 159)(61 117)(62 118)(63 119)(64 120)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 101)(19 102)(20 103)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(61 96)(62 97)(63 98)(64 99)(65 100)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(121 142)(122 143)(123 144)(124 145)(125 146)(126 147)(127 148)(128 149)(129 150)(130 151)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 141)
(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 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 47 93 156)(2 56 94 145)(3 45 95 154)(4 54 96 143)(5 43 97 152)(6 52 98 141)(7 41 99 150)(8 50 100 159)(9 59 81 148)(10 48 82 157)(11 57 83 146)(12 46 84 155)(13 55 85 144)(14 44 86 153)(15 53 87 142)(16 42 88 151)(17 51 89 160)(18 60 90 149)(19 49 91 158)(20 58 92 147)(21 76 137 102)(22 65 138 111)(23 74 139 120)(24 63 140 109)(25 72 121 118)(26 61 122 107)(27 70 123 116)(28 79 124 105)(29 68 125 114)(30 77 126 103)(31 66 127 112)(32 75 128 101)(33 64 129 110)(34 73 130 119)(35 62 131 108)(36 71 132 117)(37 80 133 106)(38 69 134 115)(39 78 135 104)(40 67 136 113)
G:=sub<Sym(160)| (1,16)(2,7)(3,18)(4,9)(5,20)(6,11)(8,13)(10,15)(12,17)(14,19)(21,54)(22,45)(23,56)(24,47)(25,58)(26,49)(27,60)(28,51)(29,42)(30,53)(31,44)(32,55)(33,46)(34,57)(35,48)(36,59)(37,50)(38,41)(39,52)(40,43)(61,66)(62,77)(63,68)(64,79)(65,70)(67,72)(69,74)(71,76)(73,78)(75,80)(81,96)(82,87)(83,98)(84,89)(85,100)(86,91)(88,93)(90,95)(92,97)(94,99)(101,106)(102,117)(103,108)(104,119)(105,110)(107,112)(109,114)(111,116)(113,118)(115,120)(121,147)(122,158)(123,149)(124,160)(125,151)(126,142)(127,153)(128,144)(129,155)(130,146)(131,157)(132,148)(133,159)(134,150)(135,141)(136,152)(137,143)(138,154)(139,145)(140,156), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,81)(20,82)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,160)(42,141)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,155)(57,156)(58,157)(59,158)(60,159)(61,117)(62,118)(63,119)(64,120)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116), (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,101)(19,102)(20,103)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(61,96)(62,97)(63,98)(64,99)(65,100)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,141), (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(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), (1,47,93,156)(2,56,94,145)(3,45,95,154)(4,54,96,143)(5,43,97,152)(6,52,98,141)(7,41,99,150)(8,50,100,159)(9,59,81,148)(10,48,82,157)(11,57,83,146)(12,46,84,155)(13,55,85,144)(14,44,86,153)(15,53,87,142)(16,42,88,151)(17,51,89,160)(18,60,90,149)(19,49,91,158)(20,58,92,147)(21,76,137,102)(22,65,138,111)(23,74,139,120)(24,63,140,109)(25,72,121,118)(26,61,122,107)(27,70,123,116)(28,79,124,105)(29,68,125,114)(30,77,126,103)(31,66,127,112)(32,75,128,101)(33,64,129,110)(34,73,130,119)(35,62,131,108)(36,71,132,117)(37,80,133,106)(38,69,134,115)(39,78,135,104)(40,67,136,113)>;
G:=Group( (1,16)(2,7)(3,18)(4,9)(5,20)(6,11)(8,13)(10,15)(12,17)(14,19)(21,54)(22,45)(23,56)(24,47)(25,58)(26,49)(27,60)(28,51)(29,42)(30,53)(31,44)(32,55)(33,46)(34,57)(35,48)(36,59)(37,50)(38,41)(39,52)(40,43)(61,66)(62,77)(63,68)(64,79)(65,70)(67,72)(69,74)(71,76)(73,78)(75,80)(81,96)(82,87)(83,98)(84,89)(85,100)(86,91)(88,93)(90,95)(92,97)(94,99)(101,106)(102,117)(103,108)(104,119)(105,110)(107,112)(109,114)(111,116)(113,118)(115,120)(121,147)(122,158)(123,149)(124,160)(125,151)(126,142)(127,153)(128,144)(129,155)(130,146)(131,157)(132,148)(133,159)(134,150)(135,141)(136,152)(137,143)(138,154)(139,145)(140,156), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,81)(20,82)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,160)(42,141)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,155)(57,156)(58,157)(59,158)(60,159)(61,117)(62,118)(63,119)(64,120)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116), (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,101)(19,102)(20,103)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(61,96)(62,97)(63,98)(64,99)(65,100)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,141), (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(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), (1,47,93,156)(2,56,94,145)(3,45,95,154)(4,54,96,143)(5,43,97,152)(6,52,98,141)(7,41,99,150)(8,50,100,159)(9,59,81,148)(10,48,82,157)(11,57,83,146)(12,46,84,155)(13,55,85,144)(14,44,86,153)(15,53,87,142)(16,42,88,151)(17,51,89,160)(18,60,90,149)(19,49,91,158)(20,58,92,147)(21,76,137,102)(22,65,138,111)(23,74,139,120)(24,63,140,109)(25,72,121,118)(26,61,122,107)(27,70,123,116)(28,79,124,105)(29,68,125,114)(30,77,126,103)(31,66,127,112)(32,75,128,101)(33,64,129,110)(34,73,130,119)(35,62,131,108)(36,71,132,117)(37,80,133,106)(38,69,134,115)(39,78,135,104)(40,67,136,113) );
G=PermutationGroup([[(1,16),(2,7),(3,18),(4,9),(5,20),(6,11),(8,13),(10,15),(12,17),(14,19),(21,54),(22,45),(23,56),(24,47),(25,58),(26,49),(27,60),(28,51),(29,42),(30,53),(31,44),(32,55),(33,46),(34,57),(35,48),(36,59),(37,50),(38,41),(39,52),(40,43),(61,66),(62,77),(63,68),(64,79),(65,70),(67,72),(69,74),(71,76),(73,78),(75,80),(81,96),(82,87),(83,98),(84,89),(85,100),(86,91),(88,93),(90,95),(92,97),(94,99),(101,106),(102,117),(103,108),(104,119),(105,110),(107,112),(109,114),(111,116),(113,118),(115,120),(121,147),(122,158),(123,149),(124,160),(125,151),(126,142),(127,153),(128,144),(129,155),(130,146),(131,157),(132,148),(133,159),(134,150),(135,141),(136,152),(137,143),(138,154),(139,145),(140,156)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,97),(16,98),(17,99),(18,100),(19,81),(20,82),(21,127),(22,128),(23,129),(24,130),(25,131),(26,132),(27,133),(28,134),(29,135),(30,136),(31,137),(32,138),(33,139),(34,140),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,160),(42,141),(43,142),(44,143),(45,144),(46,145),(47,146),(48,147),(49,148),(50,149),(51,150),(52,151),(53,152),(54,153),(55,154),(56,155),(57,156),(58,157),(59,158),(60,159),(61,117),(62,118),(63,119),(64,120),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,113),(78,114),(79,115),(80,116)], [(1,104),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,111),(9,112),(10,113),(11,114),(12,115),(13,116),(14,117),(15,118),(16,119),(17,120),(18,101),(19,102),(20,103),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(61,96),(62,97),(63,98),(64,99),(65,100),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(121,142),(122,143),(123,144),(124,145),(125,146),(126,147),(127,148),(128,149),(129,150),(130,151),(131,152),(132,153),(133,154),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,141)], [(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(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)], [(1,47,93,156),(2,56,94,145),(3,45,95,154),(4,54,96,143),(5,43,97,152),(6,52,98,141),(7,41,99,150),(8,50,100,159),(9,59,81,148),(10,48,82,157),(11,57,83,146),(12,46,84,155),(13,55,85,144),(14,44,86,153),(15,53,87,142),(16,42,88,151),(17,51,89,160),(18,60,90,149),(19,49,91,158),(20,58,92,147),(21,76,137,102),(22,65,138,111),(23,74,139,120),(24,63,140,109),(25,72,121,118),(26,61,122,107),(27,70,123,116),(28,79,124,105),(29,68,125,114),(30,77,126,103),(31,66,127,112),(32,75,128,101),(33,64,129,110),(34,73,130,119),(35,62,131,108),(36,71,132,117),(37,80,133,106),(38,69,134,115),(39,78,135,104),(40,67,136,113)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | C4○D4 | D10 | Dic5 | D10 | C5⋊D4 | D4×D5 | D4⋊2D5 |
| kernel | C24.19D10 | C2×C4×Dic5 | C2×C4⋊Dic5 | C2×C23.D5 | D4×C2×C10 | D4×C10 | C2×Dic5 | C2×C20 | C22×D4 | C2×C10 | C22×C4 | C2×D4 | C24 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 4 | 1 | 8 | 4 | 4 | 2 | 4 | 2 | 8 | 4 | 16 | 4 | 4 |
Matrix representation of C24.19D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 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 |
| 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 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 7 | 0 | 0 | 0 | 0 |
| 35 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 18 | 0 | 0 | 0 | 0 |
| 2 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 18 |
| 0 | 0 | 0 | 0 | 9 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,9,0,0,0,0,0,40],[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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,35,0,0,0,0,7,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,2,0,0,0,0,18,39,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,9,0,0,0,0,18,40] >;
C24.19D10 in GAP, Magma, Sage, TeX
C_2^4._{19}D_{10} % in TeX
G:=Group("C2^4.19D10"); // GroupNames label
G:=SmallGroup(320,848);
// by ID
G=gap.SmallGroup(320,848);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=d,f^2=d*b=b*d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*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