metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.4D10, C10.72(C4×D4), (C2×C20).248D4, C23.D5⋊15C4, C22.96(D4×D5), C23.16(C4×D5), (C22×C4).25D10, C10.83(C4⋊D4), (C2×Dic5).228D4, C2.2(Dic5⋊D4), C2.1(C20.17D4), C10.37(C4.4D4), C22.51(C4○D20), (C23×C10).28C22, C5⋊6(C24.C22), C23.278(C22×D5), C10.10C42⋊11C2, C10.46(C42⋊C2), C10.12(C42⋊2C2), C2.25(Dic5⋊4D4), C2.5(D10.12D4), C22.44(D4⋊2D5), (C22×C10).320C23, (C22×C20).342C22, C2.6(C23.D10), C10.28(C22.D4), (C22×Dic5).36C22, C2.14(C23.11D10), C2.8(C4×C5⋊D4), (C2×C4×Dic5)⋊22C2, (C2×C22⋊C4).7D5, C22.124(C2×C4×D5), (C2×C10).430(C2×D4), (C2×C4).98(C5⋊D4), C22.48(C2×C5⋊D4), (C2×C23.D5).7C2, (C2×C10).75(C4○D4), (C2×C10.D4)⋊32C2, (C10×C22⋊C4).24C2, (C2×C10).207(C22×C4), (C22×C10).116(C2×C4), (C2×Dic5).104(C2×C4), SmallGroup(320,572)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.4D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=b, f2=db=bd, ab=ba, eae-1=ac=ca, ad=da, faf-1=abcd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 590 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C42, 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, C10.D4, C23.D5, C23.D5, C5×C22⋊C4, C22×Dic5, C22×C20, C23×C10, C10.10C42, C2×C4×Dic5, C2×C10.D4, C2×C23.D5, C10×C22⋊C4, C24.4D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C4×D5, C5⋊D4, C22×D5, C24.C22, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, C23.11D10, C23.D10, Dic5⋊4D4, D10.12D4, C4×C5⋊D4, C20.17D4, Dic5⋊D4, C24.4D10
►On 160 points
Generators in S160
(2 39)(4 21)(6 23)(8 25)(10 27)(12 29)(14 31)(16 33)(18 35)(20 37)(41 96)(42 127)(43 98)(44 129)(45 100)(46 131)(47 82)(48 133)(49 84)(50 135)(51 86)(52 137)(53 88)(54 139)(55 90)(56 121)(57 92)(58 123)(59 94)(60 125)(61 87)(62 138)(63 89)(64 140)(65 91)(66 122)(67 93)(68 124)(69 95)(70 126)(71 97)(72 128)(73 99)(74 130)(75 81)(76 132)(77 83)(78 134)(79 85)(80 136)(102 151)(104 153)(106 155)(108 157)(110 159)(112 141)(114 143)(116 145)(118 147)(120 149)
(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 38)(2 39)(3 40)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 150)(102 151)(103 152)(104 153)(105 154)(106 155)(107 156)(108 157)(109 158)(110 159)(111 160)(112 141)(113 142)(114 143)(115 144)(116 145)(117 146)(118 147)(119 148)(120 149)
(1 156)(2 157)(3 158)(4 159)(5 160)(6 141)(7 142)(8 143)(9 144)(10 145)(11 146)(12 147)(13 148)(14 149)(15 150)(16 151)(17 152)(18 153)(19 154)(20 155)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 100)(56 81)(57 82)(58 83)(59 84)(60 85)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(73 139)(74 140)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)
(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 83 146 48)(2 92 147 57)(3 81 148 46)(4 90 149 55)(5 99 150 44)(6 88 151 53)(7 97 152 42)(8 86 153 51)(9 95 154 60)(10 84 155 49)(11 93 156 58)(12 82 157 47)(13 91 158 56)(14 100 159 45)(15 89 160 54)(16 98 141 43)(17 87 142 52)(18 96 143 41)(19 85 144 50)(20 94 145 59)(21 140 120 64)(22 129 101 73)(23 138 102 62)(24 127 103 71)(25 136 104 80)(26 125 105 69)(27 134 106 78)(28 123 107 67)(29 132 108 76)(30 121 109 65)(31 130 110 74)(32 139 111 63)(33 128 112 72)(34 137 113 61)(35 126 114 70)(36 135 115 79)(37 124 116 68)(38 133 117 77)(39 122 118 66)(40 131 119 75)
G:=sub<Sym(160)| (2,39)(4,21)(6,23)(8,25)(10,27)(12,29)(14,31)(16,33)(18,35)(20,37)(41,96)(42,127)(43,98)(44,129)(45,100)(46,131)(47,82)(48,133)(49,84)(50,135)(51,86)(52,137)(53,88)(54,139)(55,90)(56,121)(57,92)(58,123)(59,94)(60,125)(61,87)(62,138)(63,89)(64,140)(65,91)(66,122)(67,93)(68,124)(69,95)(70,126)(71,97)(72,128)(73,99)(74,130)(75,81)(76,132)(77,83)(78,134)(79,85)(80,136)(102,151)(104,153)(106,155)(108,157)(110,159)(112,141)(114,143)(116,145)(118,147)(120,149), (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), (1,38)(2,39)(3,40)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,141)(113,142)(114,143)(115,144)(116,145)(117,146)(118,147)(119,148)(120,149), (1,156)(2,157)(3,158)(4,159)(5,160)(6,141)(7,142)(8,143)(9,144)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,151)(17,152)(18,153)(19,154)(20,155)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,81)(57,82)(58,83)(59,84)(60,85)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126), (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), (1,83,146,48)(2,92,147,57)(3,81,148,46)(4,90,149,55)(5,99,150,44)(6,88,151,53)(7,97,152,42)(8,86,153,51)(9,95,154,60)(10,84,155,49)(11,93,156,58)(12,82,157,47)(13,91,158,56)(14,100,159,45)(15,89,160,54)(16,98,141,43)(17,87,142,52)(18,96,143,41)(19,85,144,50)(20,94,145,59)(21,140,120,64)(22,129,101,73)(23,138,102,62)(24,127,103,71)(25,136,104,80)(26,125,105,69)(27,134,106,78)(28,123,107,67)(29,132,108,76)(30,121,109,65)(31,130,110,74)(32,139,111,63)(33,128,112,72)(34,137,113,61)(35,126,114,70)(36,135,115,79)(37,124,116,68)(38,133,117,77)(39,122,118,66)(40,131,119,75)>;
G:=Group( (2,39)(4,21)(6,23)(8,25)(10,27)(12,29)(14,31)(16,33)(18,35)(20,37)(41,96)(42,127)(43,98)(44,129)(45,100)(46,131)(47,82)(48,133)(49,84)(50,135)(51,86)(52,137)(53,88)(54,139)(55,90)(56,121)(57,92)(58,123)(59,94)(60,125)(61,87)(62,138)(63,89)(64,140)(65,91)(66,122)(67,93)(68,124)(69,95)(70,126)(71,97)(72,128)(73,99)(74,130)(75,81)(76,132)(77,83)(78,134)(79,85)(80,136)(102,151)(104,153)(106,155)(108,157)(110,159)(112,141)(114,143)(116,145)(118,147)(120,149), (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), (1,38)(2,39)(3,40)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,141)(113,142)(114,143)(115,144)(116,145)(117,146)(118,147)(119,148)(120,149), (1,156)(2,157)(3,158)(4,159)(5,160)(6,141)(7,142)(8,143)(9,144)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,151)(17,152)(18,153)(19,154)(20,155)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,81)(57,82)(58,83)(59,84)(60,85)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126), (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), (1,83,146,48)(2,92,147,57)(3,81,148,46)(4,90,149,55)(5,99,150,44)(6,88,151,53)(7,97,152,42)(8,86,153,51)(9,95,154,60)(10,84,155,49)(11,93,156,58)(12,82,157,47)(13,91,158,56)(14,100,159,45)(15,89,160,54)(16,98,141,43)(17,87,142,52)(18,96,143,41)(19,85,144,50)(20,94,145,59)(21,140,120,64)(22,129,101,73)(23,138,102,62)(24,127,103,71)(25,136,104,80)(26,125,105,69)(27,134,106,78)(28,123,107,67)(29,132,108,76)(30,121,109,65)(31,130,110,74)(32,139,111,63)(33,128,112,72)(34,137,113,61)(35,126,114,70)(36,135,115,79)(37,124,116,68)(38,133,117,77)(39,122,118,66)(40,131,119,75) );
G=PermutationGroup([[(2,39),(4,21),(6,23),(8,25),(10,27),(12,29),(14,31),(16,33),(18,35),(20,37),(41,96),(42,127),(43,98),(44,129),(45,100),(46,131),(47,82),(48,133),(49,84),(50,135),(51,86),(52,137),(53,88),(54,139),(55,90),(56,121),(57,92),(58,123),(59,94),(60,125),(61,87),(62,138),(63,89),(64,140),(65,91),(66,122),(67,93),(68,124),(69,95),(70,126),(71,97),(72,128),(73,99),(74,130),(75,81),(76,132),(77,83),(78,134),(79,85),(80,136),(102,151),(104,153),(106,155),(108,157),(110,159),(112,141),(114,143),(116,145),(118,147),(120,149)], [(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)], [(1,38),(2,39),(3,40),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(81,131),(82,132),(83,133),(84,134),(85,135),(86,136),(87,137),(88,138),(89,139),(90,140),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(97,127),(98,128),(99,129),(100,130),(101,150),(102,151),(103,152),(104,153),(105,154),(106,155),(107,156),(108,157),(109,158),(110,159),(111,160),(112,141),(113,142),(114,143),(115,144),(116,145),(117,146),(118,147),(119,148),(120,149)], [(1,156),(2,157),(3,158),(4,159),(5,160),(6,141),(7,142),(8,143),(9,144),(10,145),(11,146),(12,147),(13,148),(14,149),(15,150),(16,151),(17,152),(18,153),(19,154),(20,155),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(49,94),(50,95),(51,96),(52,97),(53,98),(54,99),(55,100),(56,81),(57,82),(58,83),(59,84),(60,85),(61,127),(62,128),(63,129),(64,130),(65,131),(66,132),(67,133),(68,134),(69,135),(70,136),(71,137),(72,138),(73,139),(74,140),(75,121),(76,122),(77,123),(78,124),(79,125),(80,126)], [(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)], [(1,83,146,48),(2,92,147,57),(3,81,148,46),(4,90,149,55),(5,99,150,44),(6,88,151,53),(7,97,152,42),(8,86,153,51),(9,95,154,60),(10,84,155,49),(11,93,156,58),(12,82,157,47),(13,91,158,56),(14,100,159,45),(15,89,160,54),(16,98,141,43),(17,87,142,52),(18,96,143,41),(19,85,144,50),(20,94,145,59),(21,140,120,64),(22,129,101,73),(23,138,102,62),(24,127,103,71),(25,136,104,80),(26,125,105,69),(27,134,106,78),(28,123,107,67),(29,132,108,76),(30,121,109,65),(31,130,110,74),(32,139,111,63),(33,128,112,72),(34,137,113,61),(35,126,114,70),(36,135,115,79),(37,124,116,68),(38,133,117,77),(39,122,118,66),(40,131,119,75)]])
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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C4×D5 | C4○D20 | D4×D5 | D4⋊2D5 |
| kernel | C24.4D10 | C10.10C42 | C2×C4×Dic5 | C2×C10.D4 | C2×C23.D5 | C10×C22⋊C4 | C23.D5 | C2×Dic5 | C2×C20 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 2 | 8 | 4 | 2 | 8 | 8 | 8 | 2 | 6 |
Matrix representation of C24.4D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 21 | 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 | 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 |
| 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 11 | 33 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 6 |
| 0 | 0 | 0 | 0 | 8 | 22 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[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],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,36,11,0,0,0,0,0,33],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,0,0,19,8,0,0,0,0,6,22] >;
C24.4D10 in GAP, Magma, Sage, TeX
C_2^4._4D_{10} % in TeX
G:=Group("C2^4.4D10"); // GroupNames label
G:=SmallGroup(320,572);
// by ID
G=gap.SmallGroup(320,572);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,64,926,219,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=b,f^2=d*b=b*d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*b*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