metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.7D10, C23.5Dic10, (C2×C20).50D4, C2.5(C20⋊D4), (C2×Dic5).63D4, (C22×C4).29D10, C22.239(D4×D5), (C22×C10).13Q8, C10.11(C4⋊1D4), C5⋊2(C23.4Q8), C10.57(C22⋊Q8), C2.8(C20.48D4), C22.96(C4○D20), (C23×C10).32C22, (C22×C20).58C22, C22.46(C2×Dic10), C23.368(C22×D5), C10.10C42⋊29C2, C22.94(D4⋊2D5), (C22×C10).324C23, C2.20(D10.12D4), C2.6(C23.18D10), C10.29(C22.D4), (C22×Dic5).40C22, C2.22(Dic5.14D4), (C2×C4⋊Dic5)⋊9C2, (C2×C10).34(C2×Q8), (C2×C10.D4)⋊9C2, (C2×C10).318(C2×D4), (C2×C4).29(C5⋊D4), (C2×C22⋊C4).12D5, (C2×C10).78(C4○D4), (C10×C22⋊C4).14C2, C22.124(C2×C5⋊D4), (C2×C23.D5).11C2, SmallGroup(320,576)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.7D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=dc=cd, f2=c, 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=cde9 >
Subgroups: 614 in 186 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2×C10, 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, C23.4Q8, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C22×Dic5, C22×C20, C23×C10, C10.10C42, C2×C10.D4, C2×C4⋊Dic5, C2×C23.D5, C10×C22⋊C4, C24.7D10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C22.D4, C4⋊1D4, Dic10, C5⋊D4, C22×D5, C23.4Q8, C2×Dic10, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic5.14D4, D10.12D4, C20.48D4, C23.18D10, C20⋊D4, C24.7D10
►On 160 points
Generators in S160
(2 49)(4 51)(6 53)(8 55)(10 57)(12 59)(14 41)(16 43)(18 45)(20 47)(21 130)(23 132)(25 134)(27 136)(29 138)(31 140)(33 122)(35 124)(37 126)(39 128)(61 146)(62 101)(63 148)(64 103)(65 150)(66 105)(67 152)(68 107)(69 154)(70 109)(71 156)(72 111)(73 158)(74 113)(75 160)(76 115)(77 142)(78 117)(79 144)(80 119)(81 110)(82 157)(83 112)(84 159)(85 114)(86 141)(87 116)(88 143)(89 118)(90 145)(91 120)(92 147)(93 102)(94 149)(95 104)(96 151)(97 106)(98 153)(99 108)(100 155)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 133)(61 120)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 156)(82 157)(83 158)(84 159)(85 160)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)
(1 58)(2 59)(3 60)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 140)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 133)(35 134)(36 135)(37 136)(38 137)(39 138)(40 139)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)(79 99)(80 100)(101 157)(102 158)(103 159)(104 160)(105 141)(106 142)(107 143)(108 144)(109 145)(110 146)(111 147)(112 148)(113 149)(114 150)(115 151)(116 152)(117 153)(118 154)(119 155)(120 156)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 130)(22 131)(23 132)(24 133)(25 134)(26 135)(27 136)(28 137)(29 138)(30 139)(31 140)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)
(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)
(1 160 58 104)(2 159 59 103)(3 158 60 102)(4 157 41 101)(5 156 42 120)(6 155 43 119)(7 154 44 118)(8 153 45 117)(9 152 46 116)(10 151 47 115)(11 150 48 114)(12 149 49 113)(13 148 50 112)(14 147 51 111)(15 146 52 110)(16 145 53 109)(17 144 54 108)(18 143 55 107)(19 142 56 106)(20 141 57 105)(21 96 140 76)(22 95 121 75)(23 94 122 74)(24 93 123 73)(25 92 124 72)(26 91 125 71)(27 90 126 70)(28 89 127 69)(29 88 128 68)(30 87 129 67)(31 86 130 66)(32 85 131 65)(33 84 132 64)(34 83 133 63)(35 82 134 62)(36 81 135 61)(37 100 136 80)(38 99 137 79)(39 98 138 78)(40 97 139 77)
G:=sub<Sym(160)| (2,49)(4,51)(6,53)(8,55)(10,57)(12,59)(14,41)(16,43)(18,45)(20,47)(21,130)(23,132)(25,134)(27,136)(29,138)(31,140)(33,122)(35,124)(37,126)(39,128)(61,146)(62,101)(63,148)(64,103)(65,150)(66,105)(67,152)(68,107)(69,154)(70,109)(71,156)(72,111)(73,158)(74,113)(75,160)(76,115)(77,142)(78,117)(79,144)(80,119)(81,110)(82,157)(83,112)(84,159)(85,114)(86,141)(87,116)(88,143)(89,118)(90,145)(91,120)(92,147)(93,102)(94,149)(95,104)(96,151)(97,106)(98,153)(99,108)(100,155), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,120)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,156)(82,157)(83,158)(84,159)(85,160)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155), (1,58)(2,59)(3,60)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,140)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,133)(35,134)(36,135)(37,136)(38,137)(39,138)(40,139)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,100)(101,157)(102,158)(103,159)(104,160)(105,141)(106,142)(107,143)(108,144)(109,145)(110,146)(111,147)(112,148)(113,149)(114,150)(115,151)(116,152)(117,153)(118,154)(119,155)(120,156), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,130)(22,131)(23,132)(24,133)(25,134)(26,135)(27,136)(28,137)(29,138)(30,139)(31,140)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146), (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), (1,160,58,104)(2,159,59,103)(3,158,60,102)(4,157,41,101)(5,156,42,120)(6,155,43,119)(7,154,44,118)(8,153,45,117)(9,152,46,116)(10,151,47,115)(11,150,48,114)(12,149,49,113)(13,148,50,112)(14,147,51,111)(15,146,52,110)(16,145,53,109)(17,144,54,108)(18,143,55,107)(19,142,56,106)(20,141,57,105)(21,96,140,76)(22,95,121,75)(23,94,122,74)(24,93,123,73)(25,92,124,72)(26,91,125,71)(27,90,126,70)(28,89,127,69)(29,88,128,68)(30,87,129,67)(31,86,130,66)(32,85,131,65)(33,84,132,64)(34,83,133,63)(35,82,134,62)(36,81,135,61)(37,100,136,80)(38,99,137,79)(39,98,138,78)(40,97,139,77)>;
G:=Group( (2,49)(4,51)(6,53)(8,55)(10,57)(12,59)(14,41)(16,43)(18,45)(20,47)(21,130)(23,132)(25,134)(27,136)(29,138)(31,140)(33,122)(35,124)(37,126)(39,128)(61,146)(62,101)(63,148)(64,103)(65,150)(66,105)(67,152)(68,107)(69,154)(70,109)(71,156)(72,111)(73,158)(74,113)(75,160)(76,115)(77,142)(78,117)(79,144)(80,119)(81,110)(82,157)(83,112)(84,159)(85,114)(86,141)(87,116)(88,143)(89,118)(90,145)(91,120)(92,147)(93,102)(94,149)(95,104)(96,151)(97,106)(98,153)(99,108)(100,155), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,120)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,156)(82,157)(83,158)(84,159)(85,160)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155), (1,58)(2,59)(3,60)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,140)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,133)(35,134)(36,135)(37,136)(38,137)(39,138)(40,139)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,100)(101,157)(102,158)(103,159)(104,160)(105,141)(106,142)(107,143)(108,144)(109,145)(110,146)(111,147)(112,148)(113,149)(114,150)(115,151)(116,152)(117,153)(118,154)(119,155)(120,156), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,130)(22,131)(23,132)(24,133)(25,134)(26,135)(27,136)(28,137)(29,138)(30,139)(31,140)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146), (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), (1,160,58,104)(2,159,59,103)(3,158,60,102)(4,157,41,101)(5,156,42,120)(6,155,43,119)(7,154,44,118)(8,153,45,117)(9,152,46,116)(10,151,47,115)(11,150,48,114)(12,149,49,113)(13,148,50,112)(14,147,51,111)(15,146,52,110)(16,145,53,109)(17,144,54,108)(18,143,55,107)(19,142,56,106)(20,141,57,105)(21,96,140,76)(22,95,121,75)(23,94,122,74)(24,93,123,73)(25,92,124,72)(26,91,125,71)(27,90,126,70)(28,89,127,69)(29,88,128,68)(30,87,129,67)(31,86,130,66)(32,85,131,65)(33,84,132,64)(34,83,133,63)(35,82,134,62)(36,81,135,61)(37,100,136,80)(38,99,137,79)(39,98,138,78)(40,97,139,77) );
G=PermutationGroup([[(2,49),(4,51),(6,53),(8,55),(10,57),(12,59),(14,41),(16,43),(18,45),(20,47),(21,130),(23,132),(25,134),(27,136),(29,138),(31,140),(33,122),(35,124),(37,126),(39,128),(61,146),(62,101),(63,148),(64,103),(65,150),(66,105),(67,152),(68,107),(69,154),(70,109),(71,156),(72,111),(73,158),(74,113),(75,160),(76,115),(77,142),(78,117),(79,144),(80,119),(81,110),(82,157),(83,112),(84,159),(85,114),(86,141),(87,116),(88,143),(89,118),(90,145),(91,120),(92,147),(93,102),(94,149),(95,104),(96,151),(97,106),(98,153),(99,108),(100,155)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(41,134),(42,135),(43,136),(44,137),(45,138),(46,139),(47,140),(48,121),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,131),(59,132),(60,133),(61,120),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,113),(75,114),(76,115),(77,116),(78,117),(79,118),(80,119),(81,156),(82,157),(83,158),(84,159),(85,160),(86,141),(87,142),(88,143),(89,144),(90,145),(91,146),(92,147),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(99,154),(100,155)], [(1,58),(2,59),(3,60),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,140),(22,121),(23,122),(24,123),(25,124),(26,125),(27,126),(28,127),(29,128),(30,129),(31,130),(32,131),(33,132),(34,133),(35,134),(36,135),(37,136),(38,137),(39,138),(40,139),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92),(73,93),(74,94),(75,95),(76,96),(77,97),(78,98),(79,99),(80,100),(101,157),(102,158),(103,159),(104,160),(105,141),(106,142),(107,143),(108,144),(109,145),(110,146),(111,147),(112,148),(113,149),(114,150),(115,151),(116,152),(117,153),(118,154),(119,155),(120,156)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,130),(22,131),(23,132),(24,133),(25,134),(26,135),(27,136),(28,137),(29,138),(30,139),(31,140),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,129),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,81),(72,82),(73,83),(74,84),(75,85),(76,86),(77,87),(78,88),(79,89),(80,90),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,155),(110,156),(111,157),(112,158),(113,159),(114,160),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146)], [(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)], [(1,160,58,104),(2,159,59,103),(3,158,60,102),(4,157,41,101),(5,156,42,120),(6,155,43,119),(7,154,44,118),(8,153,45,117),(9,152,46,116),(10,151,47,115),(11,150,48,114),(12,149,49,113),(13,148,50,112),(14,147,51,111),(15,146,52,110),(16,145,53,109),(17,144,54,108),(18,143,55,107),(19,142,56,106),(20,141,57,105),(21,96,140,76),(22,95,121,75),(23,94,122,74),(24,93,123,73),(25,92,124,72),(26,91,125,71),(27,90,126,70),(28,89,127,69),(29,88,128,68),(30,87,129,67),(31,86,130,66),(32,85,131,65),(33,84,132,64),(34,83,133,63),(35,82,134,62),(36,81,135,61),(37,100,136,80),(38,99,137,79),(39,98,138,78),(40,97,139,77)]])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | Dic10 | C4○D20 | D4×D5 | D4⋊2D5 |
| kernel | C24.7D10 | C10.10C42 | C2×C10.D4 | C2×C4⋊Dic5 | C2×C23.D5 | C10×C22⋊C4 | C2×Dic5 | C2×C20 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 6 | 4 | 2 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C24.7D10 ►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 | 1 | 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 | 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 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 16 | 0 | 0 |
| 0 | 0 | 25 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,39,25,0,0,0,0,16,25,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,2,1] >;
C24.7D10 in GAP, Magma, Sage, TeX
C_2^4._7D_{10} % in TeX
G:=Group("C2^4.7D10"); // GroupNames label
G:=SmallGroup(320,576);
// by ID
G=gap.SmallGroup(320,576);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,344,254,387,12550]);
// 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=c,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*d*e^9>;
// generators/relations